LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAICN 510.84 I£6r no. 188-199 cop.SU S 16.14- Report No. 192 ON THE STRUCTURE AND AUTOMORPHISMS OF FINITE AUTOMATA by Zamir Bavel October 1, 1965 JAN 27 1 DEPARTMENT OF COMPUTER SCIENCE • UNIVERSITY OF ILLINOIS • URB The person charging this material is re- sponsible for its return on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. University of Illinois Library Report No. 192 ON THE STRUCTURE AND AUTOMORPHISMS OF FINITE AUTOMATA* by Zamir Bavel October 1, 1965 Department of Computer Science University of Illinois Urbana, Illinois * This work was submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, October, 1965 . Digitized by the Internet Archive in 2013 http://archive.org/details/onstructureautom192bave ACKNOWLEDGEMENT The author wishes to express his deep gratitude and appreciation to Professor Franz E. Hohn for the guidance, en- couragement, inspiration, professional and personal stimulation, helpful comments and advice, and the privilege of association he has given the author, and for the painstaking and conscientious critique he has given this thesis. The author wishes to express special thanks to Professor David E* Muller for the many helpful conversations, out of which came the motivation for this thesis, for his willingness and readiness to help, and for his critical assistance and collabo- ration in the past two years. The author also wishes to acknowledge and thank the Danforth Foundation for the two years of the Danforth Teacher Grant, without which this thesis and the work leading to it would have been impossible. - iii TABLE OF CONTENTS Page 0. INTRODUCTION 1 1. MINIMAL AUTOMATA 8 2. GENERATORS AND PRIMARIES Ik 3. SOURCE 19 4. HOMOMORPHISMS AND SOURCE 31 5. MINIMAL AUTOMATA, CYCLES, AND AUTOMORPHISMS 40 6. PRIMARIES AND AUTOMORPHISMS 57 7. EXTENSIONS OF AUTOMORPHISMS 62 8. ADDITIONAL RESULTS AND POSTSCRIPT 80 BIBLIOGRAPHY 87 VITA 89 - iv - Oo INTRODUCTION In this section we establish the notational conventions and list the basic definitions in the theory of automata, which we use in the following sections . We also describe the content and explain the organization of the paper in brief . The term "automaton" is to mean "finite automaton" throughout the paper . As we are not concerned with the output feature of sequential machines , we ignore both the set of outputs and the output function in the definition of an automaton Polo Definition? An automaton A is a triple (Q, I, 6), where Q is a finite set (the set of states ), I is a non-empty semigroup (the input semigroup ) , and 6 is a function (the transition function ) on Q x I into Q, which satisfies 6(q, xy) ■ 6[6(q, x), y], for all q € Q and for all x,y € Io This definition is similar to those of Fleck [5], Rabin and Scott [l], and Ginsburg [3], but with the set of outputs and the output function deleted from the latter o These three defi- nitions differ from #0.1 mainly in that we allow the set, Q, of states of the automaton A to be empty; i„eo, Q = d, 0o2o Definition s The automaton ( 6(q, x) = 6(q, y) for all - 3 - / q € Qo We shall use definitions ## o 2 and 0„3 interchangeably, with M denoting the set from which I is generatedo 0o4 o Definition ; Let A= (Q, I, 6). We denote by I* the augmented i nput semigroup of A , where the elements of I* are the empty input, €, and the elements of I, and where x€ = €x = x, for all x € I*<= The natural generalization of an automaton appears in the following definition., 0o5" De finition; A monadic algebra A is a triple (Q, M 9 6) 9 where Q is the set of elements of A, M is the set of operation indices of A 9 and 6 is the transition function 6s Q x M* — => Q of A, with the property, 6{q, xy) - 6[6(q,x) 9 y] for all q € Q and for all x,y € M*o Thus, an automaton is a finite monadic algebra; i e , a monadic algebra with a finite set, Q 9 of elements (and hence a finite set, M 9 of non-redundant operation indices )o The term "algebra" is to mean "monadic algebra" throughout this paper D 0o6o Def inition; An automaton A ■ (Q, I, 6) is said to be strongly connected if, and only if, for any p,q € Q, there exists x € I, such that 6(p, x) = q c Oo7o Definition ; Let A = (Q, I, 6) and let P c Q. Then, B = (P, I, 6*) is said to be a subautomaton of A , denoted B « A, if, and only if, for all p € P and for all x € I ♦ » 6°(p, x) € P, where 6° is the restriction of 6 to P x I. The reader will note that, for any two subautomata, B and C, B « C and C « B «&=£> B = Co For any B « A, we denote the set of states of B by Q-g, unless this set is explicitly given a different denotation,. At times, when no ambiguity may arise, we use the same symbol, 6, for the transition functions of both an automaton and its subautomaton We make use of the customary notation of set theory and, where we apply it to automata, rather than to sets, we state the sense in which the symbols are usedo Here, we need only point out that, where B is a subset of states of an automaton (Q, I, 6) - B d S f Q - B. As a result of typographical restrictions, we use the symbol "#" in three different ways: Where B is a set, #(B) denotes the cardinality of Bo We further use M #" to specify the numerical label of a result (e.g., # 3.2 refers to the label the lemma which appears as the second labeled item of section Lastly, in lemma # 5„20 we define a group operation and denote it by "# M ( , in each case, the context will make clear the rise In which this symbol is used,, We now detail the organizations and briefly describe a part of the content, of the remainder of the paper „ In section # 1, we define a minimal automaton and illustrate some of the advantages it affords „ In section # 2, we formulate the basic notion of a primary of an automaton and that of the generators of the primary We show that the set of primaries of an automaton is both unique and exhaustive, but the fundamental nature of the primary is investigated more fully in succeeding sections o In section # 3» we formulate the basic notion of the source of a subset of states and explore its use D We illustrate the characterization via the source of such notions as? strong connectedness, separated subsets of states, discrete automata, retrievable automata (see Bavel - Muller [lO]), and reflexive automata o We also indicate further uses for the source „ In section # 4, we introduce homomorphisms, endo- morphisms, isomorphisms, and automorphisms on automata and study their application to the notion of source „ Some of the results we obtain thus are these? Any endomorphism on a strongly connected automaton is an automorphism The source of a strongly connected subautomaton B is invariant under an automorphism which maps a state of B into B Two homomorphisms, which have the same image on a state, q, are identical on the subautomaton generated by q. The homomorphic image of a discrete algebra is discrete The homomorphic image of a reflexive algebra is reflexive In section # 5» we introduce the notion of a cycle (a generalization of the classical cycle) and use it to charac- terize and count all automorphisms of a minimal automaton (uioa.). We show that any such automorphism is completely determined by its action on any generator of the m a and that any two such defining cycles (of the same automorphism) are either identical or disjoint o Furthermore, we show that the order of such an automorphism in the automorphism-group of the m.ao is equal to the length of the defining cycles and that this length divides the cardinality of the set of generators of the m a We show that the set of automorphic images of any state, q, of a m a A, with the analog "#" of functional composition as the operation, forms a group which is a homomorph of the automorphism-group, G(A), of A and that, if q is a generator of A, the said group is an isomorph of G(A), We show that the order of the automorphism-group of a m,a., A, divides the cardinality of the set of generators of A and, as a corollary, we obtain the following generalization of a theorem of Fleck [5]: The order of the automorphism-group of a strongly connected automaton, A, divides the number of states of A„ In section # 6, we apply the notions and results of evlous sections to the primaries of an automaton,, We show that every primary of the homomorphic image of an automaton is the image of some primary of the automaton; that the automorphic image of a primary is a primary; that the set, P(A) of auto- morphisms leaving all primaries invariant is a normal subgroup of the automorphism-group; that the order of any member of P(A) is the locnio of the lengths of the defining cycles in all the primaries of the automaton „ In section # 7» we study the nature, behavior, and number of the members of P(A) via extensions of automorphisms of one primary to another. We introduce the notion of a component , study its behavior under automorphisms 9 and use it to reduce the scope of the investigation We characterize and count all ex- tensions of an automorphism of a primary, R, to a primary, S, and thus are able to characterize and count all members of P(R U S)c We also indicate the extension of this process to the entirety of P(A) In section # 8, further results by the author, based on this paper, are briefly described and some tempting directions of further investigation are indicatedo lo MINIMAL AUTOMATA We shall need the following notational conventions for sets of successors and the few elementary results concerning them 1.1. Definition ; Let A = (Q, I, 6) be an automaton with R c Q« Then, (i) 6(R) = {6(r, x): r € R, x € I}; (ii) 6 (R) = {6(r, x): r € R, x € I*}; (ill) 6(r) = 6({r}), r € Q; (iv) 6 (r) = 6 ({r}), r € Q Q We note that 6q(R) contains the successor of each state in R by the empty input, €, while 6(R) may not. 1.2 . Lemma : Let B, C c Q, where A = (Q, I, 6)o Then, (i) 6(B U C) = 6(B) U 6(C); 6 Q (B U C) = 6 Q (B) U 6 (C) o (ii) 6(B flC)c 6(B) n 6(C); 6 Q (B fl C) £ 6 Q (B) fl 6 (C)o (iii) B £ C => 6(B) £ 6(C) and 6 Q (B) c 6q(C)o (iv) 6 [6(B)] = 6[6 Q (B)] = 6(B); 6 [6 Q (B)] = 6 Q (B)o Proof: (i) q € 6 (B U C) <=> there exists p € B U C and x € I, such that q = 6(p, x) <=> there exists p € B such that q = 6(p, x) or there exists p € C such that q = 6(p, x), for some x 6 I q € 6(B) or q € 6(C) <=> q € 6(B) U 6(C). (ii) q € 6(B n C) ==> there exist p € B n C and x € I, such that q = 6(p, x) => q € 6(B) and q € 6(C) ===> q c 6(b) n 6(c). - 8 - (iii) Let q € 6(B)o Then, there exist p 6 B and x € I, such that q ■ 6(p, x) Q But, B £ C ===> p € C ==> q € 6(C), The proofs for the second part of (i), (II), and (iii) are identical to those for the corresponding first part, with 6q replacing 6 and I* replacing I. (iv) This is immediate from the definition # 1.1. Q.E.Do The following example demonstrates that the inclusion in # 1»2 (ii) cannot be replaced by equality: Let Q = {p, q, r} , M = {x} , and 6(p, x) ■ 6(q, x) ■ 6(r, x) = r. Here, 6 ( {p} {ql ) = 6(*0 = rf, while 6(p) n 6(q) = {r} . The following example demonstrates that # lr,2 (Iv) can- not be strengthened to read: 6[6q(B)] = 6q(B) and 6[6(B)] = 6(B), Let Q = {p, q, r} 9 M ■ {x} , 6(p, x) = q, 6(q, x) = 6(r, x) = r. -V0-^£> Here, 6 Q (Q) = Q, 6[6 Q (Q)] - 6(Q) - {q 9 r} y* 6 (Q), and 6[6(Q)] - 6({q s r}) = {r} ^ 6(Q)» One of our main devices, which we define in # 1 3> below, is the minimal automaton containing a given subset, P, of - 10 - the set, Q, of states of an automaton; i.e., the automaton generated by P. 1.3. Definition ; Let P c Q, where A = (Q, I, 6)0 The minimal automaton of P, denoted by A(P), is defined by, A(P) = n {B « A: P c Q B }. When P is a singleton (P = {p} ) , we omit the set designation, def so that A(p) = A({p}), as there is no danger of ambiguity . The reader will note that both the definition and the notation in # lo3 are consistent with those of the empty auto- maton, A(szf), of definition # o 2 o It is easily seen that the intersection of subautomata is itself a subautomaton, since the successors of each member of the intersection are members of each intersected subautomaton and, hence, are themselves members of the intersection*, Thus, A(P) « A, for all P c Q 1.4. Definition ; Let P £ Q, where A = (Q, I, 6). The automaton generated by P, denoted by [6q(P)] 9 is defined by, (2 ) where 6 1 is the restriction of 6 to 6 Q (F) x I. Again, we the set designation when P is a singleton, so that, if , .-1 def r o^ = C*o F - since the closure of 6° is guaranteed by the definition., It is also clear that P c Qr § (p)]o 1o5q Theorem ; For all P c Q 5 A(P) = [6 (P)] o Proof ; The comment preceding the theorem implies that A(P) « [6 (P)], by ## 1.3 and 1.2 (ill). Conversely, A(P) « A =^> 6 (P) £ Q A(p) -* [6 (P)] « A(P) U QoE.Do In view of # lo5 9 we shall use definitions ## 1.3 an( i lo4 interchangeably o Some of the utility of the concept of minimal automata may be seen from the (trivial) fact that any subautomaton of A is the union of the minimal automata of its states . As we intend to use this fact implicitly, we shall dignify it by stating it as a theorem, but its proof is trivial and, hence 9 omittedo lo6o Theorem ; B « A ■-&• B = U {A(q); q € Q B K W/ As we shall make use of the following two known results (for example, see [3]? P° 30) concerning strongly connected automata, we take this opportunity to further illustrate the (3) The union of subautomata (P is I, 6 i ), i € {1, 2, o , n}, is taken to be the triple (p 9 I, 6°), with P = .U-Pj. and 6 ° is the restriction of 6 to 'P x I. - 12 - utility of the minimal automaton, which we employ in our proofs of these resultso lo7o Theorem ; Every automaton has a strongly connected subautomaton. Proof ; Any one-state automaton is clearly strongly connected^ Suppose the theorem true for all k-state automata, where k < n, and let A = (Q, I, 6) be an n-state automaton with q as one of its states <, If A(q) is not strongly connected, then there exists p € 6(q) such that, for all x € I*, 6(p, x) t q. But, A(p) « A(q) and q £ 6 Q (p) => #[Q A(p )] < #[Q A (q)^° Hence, by the inductive hypothesis, A(p) has a strongly connected subautomaton and, therefore, so does Ao Thus, the theorem is proved by induction . QoEoD. An automaton B is said to be a proper subautomaton of an automaton A if, and only if, B « A, B / A, and B 4 A((zO° 1 8„ Theorem ; An automaton is strongly connected if, and only if, it has no proper subautomaton. Proof; Suppose A = (Q, I, 6) has no proper subautoma- Then, for all q € Q, A = A(q) and, hence, for all p,q € Q, there exists x € I*, such that p = 6(q, x)„ Thus, A is 3trongly connected. Conversely, if A is strongly connected and A(^) / B = - 13 - (P, I, 6) « A, then, for all q € Q and for all p € P, there exists x € I, such that 6(p, x) = q« Hence, q € 6 (p) £ 6q(P) Q £ &o( p ) :sss> A ..., ?n } f and le t q € Q. P generates A — * A - J^-Afc^) => qeJj^Cp^ => 6 ( Pj ), for some j € {l, 2, ..., n} => 6 Q (q) £ 6 [6 (pj)], by # 1.2 (ill). Hence, by # 1.2 (iv), 6 Q (q) £ 6 (p j) and ' thus > A(q) « A(pj). Q.E.D, - 14 - - 15 - It should be noted that p. (in the proof of the lemma # 2 2) is not necessarily unique; ioe , there may exist p, € P, distinct from p., such that A(q) « A(p k )° We illustrate the use of a generating set by showing that the definition of an abelian automaton is equivalent to a weaker statement . An automaton A = (Q, I, 6) is said to be abelian (see, for example, [5]) if» and only if, for all q € Q and for all x,y € I, 6(q, xy) = 6(q, yx)o 2o3o Theorem ; An automaton is abelian if, and only if, for any generating set P of A, 6(p, xy) = 6(p, yx), for all p € P and for all x,y € L Proof ; Let P be any generating set of A = (Q, I, 6), and let q € Q Then, by # 2 2, there exists p € P, such that q € 6q(p) =■> there exists z € I*, such that q = 6(p, z) => for all x,y € I, 6(q, xy) = 6[6(p, z), xy] = 6(p, zxy) = 6[p, z(xy)] = 6[p, (xy)z] ■ 6[p, x(yz)] = 6[p, (yz)x] = &[6(p, yz), x] = 6[6(p, zy) 9 x] = 6[6(p, z), yx] = 6(q, yx) c Hence, A is abelian „ The converse is trivial • QoE»De 2o4o Definition ; A subautomaton B of an automaton A = (Q, I, 6) is said to be a primary of A if, and only if, the following two conditions are satisfied; (i) there exists q € Q, such that B = A(q); (ii) for all p € Q, B « A(p) => B = A(p)„ - 16 Thus, a primary is a maximal subautomaton generated by a single state . We now show that the set of primaries of an automaton is both exhaustive and unique . 2 g, primary Decomposition Theorem ; Let W= {P 19 ...,P n } be the set of (distinct) primaries of an automaton A = (Q, I, 6). Then, (i) A = J^; (ii) A 4 U Pu for any j € {1, 2, ooo, n} , i^j Proof ; (i) Let q Q € Q, and suppose that, for all x € I and for all q € Q - 6 (q ), 6(q, x) * ^ Then, by # 2^, A(q Q ) is a primary of A. Thus, suppose that, for some x € I and for some q-L € Q - 6 (q Q ), 6(q lf x) = q ° Then, 6 (q ) £ 6 Q (q 1 ) =*> #[« (q )]<#Cfi ^i )] - If A(q i 3 is not a primary ° f As there exist x € I and q 2 € Q - 6 Q (q 1 ), such that 6(q 2> x) = q x . Then, 6 Q (q ) % 6 Q (q 1 ) J 6 Q (q 2 ) «*> #[6 (q Q )] < #C« <*L>3 < # 6 (<1 2^ Since Q is finite, there exists q m € Q such that, q € 6 Q (q m ) and 6(r, x) + q m , for all r € Q - & (q m ) and for all x € I. Hence, for all s € Q, A(q m ) « A(s) -* A(q m ) = A(s); i.e., A(q m ) is a primary of A and, hence, is one of the P i * s in W» Thus, A = Jx*!' (ii) Let B = U P 19 and let P 1 = A(q), for some j € i^j X J {1, ..., n}. We show that q £ Q B o Suppose q € Q B „ Then, q € Qp for some i 4 J. Let P 1 = A(p). Then, q € Q Pi — * by # 1.2 (ill), 6 Q (q) C 6 (Q Pl ) = 6 (p) — *> A(q) « A(p) — > by # 2.4 (ii), A(q) = A(p) => ? i = P., contradicting the hypothesis on W, Thus, (11) is proved. QoEcDo _ 17 - We note that the same primary may have several gener- ators, since A(p) = A(q) =?^> p = q Also, the same state may belong to more than one primary; i e , distinct primaries may have a non-empty intersection 2 060 Corollary ; Let BP = {P^ ; i = 1, 2, 000 , n} be the set of primaries of an automaton A, and let R = {r-,, . «, , r^} be a minimal generating set of Au Then, n = k Proof ; For each V^ € IP, let P i = A(p i ) D Then, by # 2o5 (i)» since {p-i, p ? , 000, p } is a generating set, k < n, and by # 2„ 5 (li), since each A(t } ) « A( Pl ) for some 1, n < k. Q.E.D. 2q7q Definition ; The rank of an automaton A, denoted by r(A), is defined as the number of distinct primaries of A (or the cardinality of any minimal generating set of A) The primaries of an automaton are its natural major subdivisions o Many questions concerning automata may be answered, either largely or entirely, by investigating the primaries of the automata (as is done in following sections of this investigation) „ As an immediate illustration, we make use of primaries to charac- terize a retrievable automaton ([lO], [ll], [12], and [13]) as one whose primaries are strongly connected (and hence disjoint )„ An automaton A = (Q, I, 6) is said to be retrievable if, and only if 9 for each x € I and for each q € Q 9 there exists a corresponding y € I, such that 6(q, xy) = q Q - 18 - 2o8o Theorem ; An automaton is retrievable if, and only if, its primaries are strongly connected. Proof ; Suppose that all primaries of an automaton, A, are strongly connected, and let q € Q, x € L By # 2 5> there exists a primary, A(p), of A, such that q € 6 (p)„ Since A(p) is strongly connected, there exists y € I, such that 6[6(q, x), y] = 6(q, xy) = q, since q € 6 (p) o Hence, 6(q, x) € 6q(p) and, therefore, A is retrievable,. Conversely, let A be retrievable, let A(p) be a primary of A, and let q € 6q(p) A is retrievable => by § 1.2 (iv), 6 (q) c 6 [6 (p)] = 6 Q (p) =^> A(q) « A(p) =^> by # 2A (ii), A(q) = A(p) =*> by # 1 8 9 A(p) is strongly connected. QoEoDo 3q SOURCE Another of the elementary building blocks of automata (and, indeed, of abstract algebras) is the "source" » Intuitively, one may regard the source as a generalization of the notion of predecessors of a state of an automaton (alt„, an element of an algebra) . 3,1. Definition : Let B = (P, I, 6) « A = (Q, I, 6), and let R c p A state p € P is said to be an R- source in B if, and only if, there exists x € I, such that 6(p, x) € R. The source of R in B 9 denoted by s(R; B), is defined by, s(R; B) = {p € P: 6(p, x) € R, for some x € I*} c The pure source of R in B, denoted by s~(R; B), is defined by, s"(R; B) = {p € P: 6(p, x) € R, for some x € 1} „ The pure source of R may, but need not, include all elements of R The source of R s however, includes all elements of Ro When the automaton, A, in which the source is taken, is evident and no ambiguity may arise, we may use s(R) for s(R; A), and s~(R) for s~(R; A) We note that s~(R; B) is the set of all Resources in B, while s(R; B) = s~(R; B) U R, since s(R; B) allows predecessors by the empty input , It is, then, trivially true that, 19 - 20 - s~(R; B) c S (R; B), and that R c s{R; B)o The reverse inclusions, however , do not hold, as can be seen from the following example: Q<~ n x r~\ x,y /~ n} (iv) -s(Bj) c s(-Bj) 9 for all j e (v) s(s( o o osCB^o o o )) = sCBjl), for all io (vi) 6 (B i ) n Bj ■ 4 <^> B 1 D s(Bj) - tf? for all i,j (vii) Bi £ Bj —* s(B 1 ) £ s(Bj) 9 for all i,j Proof ; (i) q € s^ U B.) ^^ there exists x € I*, such that 6(q, x) € B^ U B, <^^=> 6(q, x ) € Bj_ or 6(q, x) € B* ^* q € s(B i ) or q € s(Bj) <£=£> q € s ^ ) U s(Bj). The result, then, follows by Induction „ - 22 - (ii) q € s(B* n B«) aa * > there exists x € I*, such that 6(q, x) € B ± fl Bj => 6(q, x) € B i and 6(q, x) € Bj => q € s(B i ) and q € s(Bj) «*> q € s(B j _) fl s(Bj)* The result, then, follows by induction (iii) q € s^) - s(Bj =«*> 6 Q (q) fl Bj = ^ and there exists x € I*, such that 6(q, x) € Bj_ => 6(q y x) € B i - Bj =^ q 6 s^ - Bj)o (iv) This result follows from (iii) when B 1 " Qo (v) Trivially, s(s(B i )) = s(B 1 )o The result, then, follows by induction o (vi) This result follows directly from the definition (vii) This result follows directly from the definition, QoEoDo It is easily seen that the inclusion in each of (ii), (iii), and (iv) of # 3-2 may not be replaced by an equality „ 3q3q Lem ma: Let C = (R, I, 6) « B « A. Then, (1) R £ s-(B; B). (ii) s~(R; B) - s(R; B) » s[6 (Rh b] = s[6(R); B] „ (iii) 6[s-(R; B)] = 6[s(R; B)]. (iv) s[6 (R)j B] c 6 [s(R; B)]. Proof : Let r € R u Then, C « B =s> 6(r) c R =£> there exists x € I, such that 6(r, x) € R —$> r € s~(R; B) =s> H £ 3 - (R; B) v and (i) is proved. The first equality in (ii) follows from (i) and the finitlon # 3,1, since s(R; B) = s"(R; B) U R., The second - 23 - equality follows from the fact that C « B =«=> R = 6 (R) o To prove the third equality, we note that, by # 3„2 (vii), 6(R) c 6 (R) ^> s[6(R); B] £ s[6 (R); B] Conversely, s(R; B) £ s[6(R); B] and hence 9 by the second equality of (ii), s[6q(R); b] £ s[6{R); B] o The two inclusions prove the third equality, which concludes the proof of (ii) The equality in (iii) is immediate from the first equality of (ii) The inclusion in (iv) is immediate from the second equality in (ii) and definition #1,1, QoEoDo We note that the inclusion in (iv) cannot be strengthen- ed to an equality, even in the two cases where B on the one hand, or (H, I, 6) on the other, is a primary. The former is evident from the first of the two examples preceding # 3°2 To demonstrate the latter case, let A = (Q, I, 6), with Q - {P s q» r, s} 9 M = {x, y}, 6(r, x) = 6(r, y) ■ 6(s, x) = 6{s, y) = 6(p, x) ■ s, and 6(p, y) = 6(q, x) = 6(q, y) = q D Let R = {r, s} 9 and let B - Ao Then, s[6 (R); B] = {p, r, s}, while 6 [s(R; B)] = {p, q, r, s} o - Zk - It is of interest to note that the source and the minimal automaton of a state, q, are complementary opposites, in that the former contains all predecessors of q, but not neces- sarily its successors, while the latter contains all successors of q, but not necessarily its predecessors . Often, a combination of the two concepts suffices to characterize a property of automa- ta, such as: A[s(q)] is the union of all primaries in which q is a state As another illustration, we present a necessary condition for an automaton to be abelian 3q^° Theorem ; In an abelian automaton, disjoint sub- automata have disjoint sources o Proof ; Let B « A, C « A, and Q B n Q c = 0% and suppose there exists q € s(B) D s(C)o Then, there exist x,y € I*, such that 6(q, x) € Q B and 6(q, y) € Q Q o But then, B « A => S(q, xy) = 6[6(q, x), y] € Q B , and C « A =*> 6(q, yx) € Q C o But, Q B n Q c = szf => 6(q, xy) ^ 6(q, yx) and, hence, A is not abelian Thus, if A is abelian, then Q B n Q c = <6 =*> s(B) n s(C) = 4. Q.E0D0 In terms of the notation of section #1, (P, I, 6) « A if, and only if, 6 (P) = p We now characterize subautomata by means of the source « 3.5. Theorem; For all P c Q, where A = (Q, I, 6), P if, and only if, s({q}) n P ± 6 =0 q € P, for any qfQ. 25 Proof ; Suppose s({q}) n P 4 4 => q € P. By # 3<>2 (vi), s({q}) nP^ <=*> {q} n 6 (P) 7* rf «=> q € 6q(P) Hence, q e 6 (P) ^=> q € P and, thus, 6 Q (P) £ p c Since, by # 1.1, the reverse Inclusion is always true, 6 Q (P) = P» Conversely, let 6q(P) = P, and suppose s({q}) D P 4 d* Then, by # 3o2 (vi ) , {q} n 6 Q (P) ¥■ 4 and, hence, q 6 6 Q (P) = P, QoE.D. In terms of the notation of section # 1, a subautomaton, B = (P, I, 6), of A is strongly connected if, and only if, for all non-empty R £ p $ 6(R) - Po We now characterize strongly connected subautomata by means of the source 3q6o Theorem; Let B - (P, I, 6) « Ao Then, B is strongly connected if, and only if, for all non-empty R c p, s(R; B) - Po Proof ; Suppose, for all non-empty R c p, s(R; B) = P, and let p,q € Po Then, {p} c s({p}; B) = P = s({q}; B) => there exists x € I*, such that 6(p, x) = q Hence, B is strongly connected „ Conversely, suppose B is strongly connected, and let R be a non-empty subset of Po Then, for any p € P, there exists x € I, such that 6(p, x) € R => P £ s(R; B) ==> P = s(R; B)o Q.E.D. - 26 - We note that theorems ## 3° 5 and 3.6 generalize to monadic algebras without change in either statement or proof . Trivially, since Q = s(Q), the set of states of an automaton is the union of the sources of all its subautomata* We now show that the strongly connected subautomata are sufficient for this purpose . 3o7o Theorem ; Let B-j_, B£» ° 00 , B n be all the strongly n connected subautomata of Ao Then, Q = U s(Q B ) Proof ; Let q € Q - iiixS^Bi) Then, by # 3 2 (i) and (vi), q i s( 8 QBi) => *oM n (J^^Bi* = ^° But » A(q) « A ==> there exists a strongly connected subautomaton C « A(q), by # 1„7 9 and, hence, Qq n ( U Q B ) = d 9 contradicting the hypothesis that B-^, B 2 9 . ••, B n are all the strongly connected subautomata of A. Hence, Q = .JJ,s(Qg ) We call a subautomaton (alt , subalgebra), B = (P,I,6), of A separated if, and only if, 6 Q (Q - P) n P = jzL We call an automaton (alt„, algebra) discrete if, and only if, each of its states (alt., elements) is a (separated) subautomaton (alt., sub- algebra) of A; i.e., for all q € Q, 6 (q) = {q) . In that case, #(Q) = r(A) and each primary of A has exactly one state (see definition # 2.7). Indeed, a discrete algebra is of little in itself; it does, however, represent the opposite of a strongly connected algebra. - 27 - We conclude this section with several characterizations of automata and algebras, which make use of the concept of source 3o8 u Theorem? Let P £ Q, where A ■ (Q, I, 6) D Then, (i) where B = (P, I, 6) « A, P = s(P) if, and only if, B is separated; (ii) P t = sCP^ for all B ± ■ (P i9 I, 6) « A if, and only if, A is retrievable (see # 2 C 8 and [lO]); (iii) P = s(P), for all P c Q 9 if, and only if, A is discrete Proof ; (i) Let B= (P, I, 6) « A, and suppose P - s(P) Then, rf = (Q - P) n P = (Q - P) n s(P) =^> &o(Q - P) n P = ^, by # 3o2 (vi), and B is separatedo Conversely, suppose B is separatedo Then, 6 (Q - P) n P = j => by # 3»2 (vi), (Q - P) n s(P) = $zf => p - p u [(q - p) n s(p)] = [p u (Q - p)] n [P u s(p)] - Q n s(p) - s(p) u (ii) Let F 1 = s(P i ), for all B i - (P l9 I, 6) « A D Let x € I, q € P^ for some i 9 and suppose 6(q, xy) ^ q, for all y € I* Then, q £ 6 [6(q, x)] and, hence 9 q 6 s(6 Q [6(q, x)]) - 6 Q [6(q, x)] But, 6 Q [6(q, x)] is the set of states of A[6(q, x)] « A, contradicting the hypothesis,, Hence, A is retrievable » Conversely, let A be retrievable , Then, by # 2.8, A is the (disjoint) union of strongly connected subautomata Q But, by # lo8, a strongly connected automaton has no proper sub- - 28 - automaton and, hence, Pj^ = s(P i ) for all subautomata B 1 = (P i , I, 6) of A. (iii) Let P = s(P), for all P £ Q, Suppose there exist q € Q and x € I, such that 6(q, x) = p 4 q. Then, q € s({p}) ==> {p} 4 s({p}), contradicting the hypothesis . Hence, for all q 6 Q, 6 Q (q) = {q} and A is discrete . The converse is trivial . Q..E.D. 3 o 9 . Remark ; Bavel and Muller ([lO], definition 1) define eight reversibility properties of monadic algebras, with k.(A), or its abbreviation k^, being the statement that algebra A possesses property i, where property i € {l, 2, 00 o, 8} is named: (1) semiretrievable, (2) retrievable s (3) unique prede- cessor, (4) onto, (5) partially reversible, (6) potentially reversible, (7) reversible, and (8) strongly connected . If we let kg(A) be the statement that algebra A is discrete, it is easily verifiable that kg => k 7 . We now define an additional reversibility property for monadic algebras and present equivalent formulations which employ the source. 3.10. Definition ; An algebra A = (Q, I, 6) is said to reflexive if, and only if, for each q € Q there exists ■ I, -ruch that 6(q, x) = q. 29 - We let k 1Q (A) be the statement that algebra A is reflexive From the following example, it is easily seen that k 1Q =f^> k lf i € {1, 2, ooo, 9): Let A = (Q, M, 6), with Q = (p, q, r} 9 M = {x, y}, 6(p s y) = p, 6(p, x) = 6(q, y) = q, and 6(q, x) = 6(r, x) ■ 6(r, y) = r Q -►fT) ►© Jx, ,llo Theorem ; The following statements are equivalent for the algebra A = (Q, M, 6): (i) A is reflexive c For all R c Q s for all B « A, (il) R £ Q B => B £ s-(R; B) (iii) R c Q B =^> R c 6[s-(R; B)]c (iv) (R, I, 6) « B =~> R £ 6[s-(R; B)]. Proof: (i) =^> (11) i Let q € R c Q B o Then, k 10 =^> there exists x € I, such that 6(q, x) = q => q € s~({q}; B) =s> q € s~(R; B) «*> R c s-(R; B) (ii) =~> (iii): R c s~(R; B) => for each r € R there exists x € I, such that 6(r, x) € R => RE 6(R)» But, R £ s~(R; B) =~> 6(R) £ 6[s"(R; B)], by # 1„2 (iii)c Hence, R £ 6(R) £ 6[s-(R; B)]. (ill) ^ (It): This is immediate, since (R, I, 6) « B —*• R c Q BO - 30 - ( lv ) ==> (i): Suppose (R, I, 6) « B => R £ 6[s"(R;B)] for all B « A, and let q € Q B . Then, since A(q) « A(q), by hypothesis, 6 Q (q) c 6[s-[6 Q (q); A(q)]] => q € 6[s~[6 (q); A(q)]] =o there exist p 6 s~[6 (q); A(q)] and x € I, such that q = 6(p, x). But, p € s-[6 (q); A(q)] => p 6 6 (q) => there exists y ^10' i° e °t a retrievable algebra is reflexive. Thus, we may amplify a conse- quence of theorem 1 in [10] to read: The following graph shows the implication relations between pairs of properties kp.. M kiQ: Two of the reversibility properties, k 2 and kg, mentioned in # 3.9, were characterized in ## 3.8 (ii) and 3.6, respectively, by means of the source. The remaining six properties may be similarly characterized, using the source as it applies to particular members of I; i.e., s x (P; B) = {q € Q B : 6(q, x n ) 6 P, n is an integer > 0}, and s x ~(P; B) = {q € Q B : 6(q, x ) € P, n is an integer > 0}. However, we omit the details, as thl s would constitute too much of a digression. 4„ H0M0M0RPHI3M3 AND SOURCE 4olo Definition ; Let A ■ (Q, I, 6) and B = (Q% I 9 6 e ) be two automata „ By a function f: A — h > B, on A to B is meant a function fs Q — > Q° o A function, f: A — > B, (5) is called a homomorphism if 9 and only if, for all q € Q and for all x € I, f[6(q, x)] = 6°[f(q), x] , By the homomorphic image , f(A), of A under the homomorphism f: A -— i > B is meant the subautomaton f(A) = (f(Q) s I, 6 8 ) « B, where f(Q) is the set of images of Q under f„ Following the common practice, we call a homomorphism, f : A — -> B, an endomorphism of A, when B - A; an isomorphism of A 9 when f is monic (i e , one-to-one) and epic (i„eo, onto); an automorphism of A, when f is an iso- morphism of A onto Ao By G(A) we denote the group of auto - morphisms of Ao k o 2 o Remark g The homomorphic image, f(A), of an automa- ton A - (Q, I, 6) is, indeed 9 a subautomaton; i e , f(Q) is closed under 6°o For, if p € f(Q), then there exists q € Q, such that p ■ f(q) and, hence, for any x € I, 6 9 (p s x) = 6 B [f(q), x] = f[6(q, x)] € f(Q)„ The set of all automorphisms of A is (trivially) a Parts of this definition parallel those of Fleck [5], Weeg [7], and Gluschkov [13] » (5) Fleck [5] and Weeg [7] call a homomorphism an "operation preserving function w - 31 . - 32 - (6 ) group under functional composition. In this investigation, we are primarily concerned with the automorphisms of an automaton. Nevertheless, when we think it appropriate, we present results of a more general nature, con- cerning homomorphisms, endomorphisms, and isomorphisms of automata 4.3. Lemma ; Let f be a homomorphism on A, let B « A, and let P c Q_. Then, (i) f[s-(P; B)] £ s-[f(P); f(B)]; (ii) f[s(P; B)] c s[f(P); f(B)]. Proof ; Let q € s~(P; B). Then, there exists x € I, such that 6(q, x) € P =s> f[6(q, x)] = fi'Cffq), x] € f(P) c f(Q B ) => f(q) € s-[f(P); f(B)] =*> f[ S -(P; B)] £ s"[f (P); f(B)]. This proves (i). Using (i) and s(P; B) = P U s"(P; B) we have, f[s(P; B)] = f(P) u f[s~(P; B)] £ f(P) U s"[f(P); f(B)] = s[f(P); f(B)]. This proves (ii). Q.E.D. We combine lemma # 4.3 and the characterization of strongly connected subautomata in theorem # 3.6 to prove; (6) fact has been noted by Fleck ([5], theorem 1). Weeg ([7], leorem A) attributes to Fleck the result; The set of all omorphisms on A is a group. However, as there is a dis- crepancy between the range and domain 6f composed endomorphisms, result does not hold, except when A is strongly connected (see # 4.8, below). - 33 - 4,4, Theorem ; The homomorphic image of a strongly- connected subautomaton is strongly connected «, Proof; Let B = (P, I 9 6) « A be strongly connected, let f be a homomorphism on A 9 and let 4 4 R £ f(F)o Then, for some non-empty T £ P, f (T) = Ro By # 3»6 9 B is strongly connected «**> P = s(T; B) — => f (P) = f[s(T; B)] £ s[f (T); f(B)], by # 4 3o But, s[f (T); f(B)] - s[H; f (B)] £ f(p) ==s> f(p) = s[R; f(B)] ^^> f(B) is strongly connected, by # 3o6, QoE.Do It should be noted that theorem # 4 C 4 may be proved without making use of the source We chose the more topical proof o *K5„ Theorem; Let R c p, where B = (P, I, 6) « A, and let h be an isomorphism on A. Then, (i) h[s(Hs B>] - s[h(R); h(B)]; (ii) h[s(R; B) - R] - s[h(R); h(B)] - h(R)„ Proof ; (i) By lemma # 4 <; 3 9 h[s(R; B)] £ s[h(R); h(B)] and it remains to show the converse o Let q 9 € s[h(R); h(B)]° Then, there exists x € I* and there exists p° € h(R) P such that 6 9 (q 9 9 x) = p 9 Since h is monic, let h" 1 {q°) ■ q and h =1 (p 9 ) = p D Then, p 9 = h(p) = 6°[h(q) 9 x] ■ h[6(q, x)] =s— > p = ${q 9 x ), where p € R and, hence, q € s(R; B)» Thus, q 9 € h[s(R; B)] and, hence, s[h(R); h(B)] c h[s(Rs B)]„ - 3k - (ii) It is an elementary property of functions that, for any function f and any sets C and D, f _1 [C - D] = f _1 [c] - f~ 1 [D] (for example, see Kelly [l^], theorem 0.7). Since h is monic, let f = h -1 , let C = s(R; B), and let D = R, Then, h[s(R; B) - R] = h[s(R; B)] - h(R) = s[h(R); h(B)] - h(R), by (i). Q.E.D. k . 6 . Remark ; In general, the equalities in # ^-.5 (i) and (ii) do not hold if h is just a homomorphism. Proof ; Let B = (Q, I, 6), with Q = {p, q, r, s} , M = {x}, 6(p, x) = r, and 6(r, x) = 6(q, x) = 6(s, x) = q. Let h be defined by, h(p) = r, h(r) ■ h(q) = q, and h(s) = s. Then, h is an endomorphism of B and h(B) = ({q, r, s} , I, 6) Now, where R = {r} , s(R; B) = {p, r} , h[s(R; B)] = {r, q} , s[h(R); h(B)] = s[{q}; h(B)] = {q, r, s} , h[s(R; B) - R] = {r} , and s[h(R); h(B)] - h(R) = {r, s}. Hence, h[s(R; B)] £ s[h(R); h(B)], and h[s(R; B) - R] ^ s[h(R); h(B)] - h(R). Q.E.D - 35 - ^o7q Theorems Let f s A — > A be an endomorphism, let B « A be strongly connected, and let q € Q B be such that f(q) € Q B o Then, f (B) - Bo Proofs Since f(q) € Q B n f(Q B ) 9 the subautomaton B fl f(B) is non-empty o But 9 by # ^K^, f (B) is strongly con- nected and 9 by # 1 8 9 neither B nor f(B) have a proper sub- automaton o Hence s B D f (B) ■ B ■ f (B)o Q.E.D, JK8_o _Co r o 1 1 ary % any endomorphism, f 9 on a strongly connected automaton is an automorphism,. Proof g By # k ? 9 f is epic and 9 since the set of states of an automaton is finite «, f is monlc QcEoDc ^o9q Corollary; Let h € G(A) and suppose that, for some strongly connected B « A and for some q € Q B , h(q) € Qg Then, h[s{Q B )] - s(Q B )o Proofs By # k«7 9 h(B) - B; i e , hfQg) - Q B „ Hence, by # 4.5 9 h[s{Q B )] - s[h(Q B )] - s(Q B ) QoE Do 4ol0c. Remark; The conclusion of # ^-o9 does not follow if any of the following conditions applies; (i) h is a non-monic homomorphism, (ii) B is not strongly connected, (iii) the last condition in the corollary is changed to read, "for some q in - 36 - s(Q„) - Q n , h(q) € s(Qo) - On-" «B« A 9 = (Q e , I 9 6°) be homomorphisms, such that f(q) = g(q), for some q € Qc Then, f = g on A(q)o If, furthermore, q 6 s(Q~), where B « A is strongly connected, then f = g on Bo Proofs Let p € 6 (q), say, p = 6(q, x) D Then, f(p) = f[6(q 9 *)] - 6 3 [f(q), x] - 6»[g(q), x] = g[6(q, x)] = g(p). The second part of the theorem is a trivial consequense of the first o Q.EoDo 4ol2o Corollary ; Let f be an endomorphism of A = (Q, I, 6) such that, for some q € Q 9 f(q) = q D Then, f is the identity on A(q)o If, furthermore, q € s(Q B ), where B « A is strongly connected, then f is the identity on B„ Proof; The corollary is immediate from # 4„11, with g taken as the identity automorphism on Ao Q.EoDo 7) By # 4 8, f and g are, then, automorphisms of A, - 38 - 4.1?. Remark ; The conditions of # 4 12 are not suf- ficient for f to be the identity on s(Q B ), even if f € G(A). Proofs Consider the automaton A = (Q, I, 6), with Q = (P. 1> r, s, t}; M = {x, y};6(p 9 x) - 6(p 9 y) = 6(r, x) = 6(r, y) = 6(q 9 x) = 6(t, x) = r 9 6(q, y) = 6(t, y) = 6(s, x) = 6(s, y) = y; h(p) = p, h(r) = r, h(s) = s 9 h(q) = t, and h(t)=q D x/ -Av x 9 y ^A^-^A^ r " r) .si )x 9 y ,3 \ **\ Z*^ X ^' Here, p 6 s({r}) and h(p) = p 9 where h € G(A)« Yet 9 q € s({r}) and h(q) = t ^ q D QoE.Do We conclude this section by showing that both of the reversibility properties, kg and k 10 9 defined in section # 3» are preserved under homomorphlsm (the interested reader will find a discussion of the homomorphic images of algebras possessing one of the remaining eight reversibility properties in [10])„ *Kl4. Theorem : The homomorphic image of a discrete algebra is discrete. Proof : Let A «= (Q, I, 6) be a discrete algebra, let fl A — > A° = (Q J , I, 6 1 ) be a homomorphlsm, let p* € h(Q), - 39 - and let q 9 € s[{p 9 }; h(A)]„ Then, there exists x € I*, such that 6 c (q B , x) = p° Let q and p be any states in Q, such that h{q) = q° and h(p) = p* u Then, s({q}) = {q} => 6(q, x) = q —> q' = h(q) - h[6(q, x)] = 6*[h(q), x] = 6 • (q * , x) s p° s^> s[{p 9 }; h(A)] ■ {p"} — > h(A) is discrete. Q.E.D, 4o15q Remark ; In contrast to # 4 l4, the homomorphic image of a separated subautomaton need not be separated, as is seen from the following example,. Let Q = {p, q, r} ; M = {x} ; 6(p, x) = p, 6(q, x) a 6(r, x) = r; f(p) = f(r) = r, and f(q)=q< i p) jx fq^- m- r) )x Here, f is a homomorphism and B ■ ( {p} , I 9 6) « A is sepa- rated o Yet y f(B) = ({r} 9 I, 6), which is not separatedo 4ol6 o Theorem ; The homomorphic image of a reflexive algebra is reflexive „ Proofs Let A = (Q 9 I, 6) be reflexive, let fs A — > A be a homomorphism, and let q 9 € f(Q)o Then, there exists q € Q, such that f(q) ■ q\ and there exists x € I, such that 6(q, x) = q Hence, q 9 = f(q) - f[6(q, x)] = 6 9 [f(q), x] = 5 s (q°, x) and, thus, f(A) is reflexive Q QoEoDo 5. MINIMAL AUTOMATA. CYCLES . AND AUTOMORPHISMS In this section we characterize automorphisms of ( 8 ) minimal automata with the aid of a new 1, tool, — the cycle 5.1. Lemma ; Let f be a homomorphism on A = (Q, I, 6) to A e = (Q\ I, 6 9 )o Then, for each q € Q, f[A(q)] = A(f(q)). Proof ; Since f[6 n (q)] = f[ U „6(q, x)] - U „f[6(q, x)] — u x€I x€I* = x U i# 6 f [f(q), x] = 6j[f(q)], we have f[A(q)] = (f[6 (q)], I, 6*) = (6^[f(q)], I, 6°) = A(f(q))o Q.E.D. 5«2o Theorem ; Let h € G(A) and let h(q) € 6 Q (q) 9 for some state q of A* Then, h[A(q)] = A[h(q)] = A(q). ' Proof ; The first equality is immediate from lemma # 5«1° Now, h(q) € 6 (q) => 6 Q [h(q)] <= 6 [_6 (q)] = 6 Q (q), by # 1.2 (iii) and (iv). Hence, h[6 Q (q)] c 6 (q), by the first equality. But, since h is monic and 6 (q) is finite, h[6 Q (q)] = 6 Q (q) => h[A(q)] = A(q) =*> A[h(q)] = A(q). Q.E.D. 5*3. Definition ; Let q be a state of A = (Q, I, 6) and let x € I. Define 6(q, x° ) ■ 6(q, €) = q. Then, 'The reader should note that the "classical " use of the term "cycle" coincides with our "circular cycle" (see # 5,7). (9) The reader may wish to compare # 5.2 with # 4.9. - 40 - 41 the X- cycle of q, denoted by C Y (q), is defined by x n C x (q) = {p € Qs p = 6(q, x ), n = 0, 1, 2, ...}. The length of the x-cycle of q is #[C (q)]o The reader will note that a cycle, as defined here, may be non-circular (see #5°?, below) „ 5 o 4 o Lemma ; #[C x (q)] ■ the least positive integer, n, for which there exists a non-negative integer k < n, such that k 6{q, x n ) = 6(q, x*) Proofs Let m be the least positive integer n for which there exists a non-negative integer k < n, such that 6(q, x n ) = 6(q s x ) c Then, by a simple induction argument, for all non-negative integers r, 6(q, x r ) € {6(q, x 1 ) : i = 0, 1, . .. coo, m - 1} and, hence, #[C_(q)] < m But, since m is the least of the n 9 s, for any s < m, 6(q, x s ) £ {6(q, x 1 ) % i = 0, 1, ooo 9 s} o Thus 9 m < #[C x (q)]o The two inequalities prove the lemma o Q.E.Do As an illustration consider the automaton with Q = {p, q, r}, M = {x 9 y}, 6(p, x) = 6(p, y) = 6(r, x) = q, 6(q, x) = 6(q s y) = r, and 6(r, y) ■ p - k2 - Here, #[C x (p)] = #{p, q, r} = 3; #[C x (q)] = #{q, r} = #[C x (r)] = 2; #[C y (p)] = #[C y (q)] = #[C y (r)] = #{p s q, r} = 3. 5o5c Theorem ; Let f be a homomorphism on A = (Q, I, 6)» Then, for any q € Q and for any x € I, (i) f[C x (q)] = C x [f(q)]; (ii) #(G x [f(q)]) < #[C x (q)]o Proof : (i) f[C x (q)] = f[{6(q, x*): n = 0, 1, 2, ...}] = {f[6(q, x n )]: n = 0, 1, 2, 000} = {6°[f(q), x n ] : n = 0, 1, 2,...} = C x [f(q)]o (ii) By # 5°^ 9 #[C x (q)] = the least positive Integer, n, such that, for some non-negative integer k < n 9 6(q, x n ) = 6(q, x k )„ Hence, 6»[f(q), x n ] = f[6(q, x 11 )] = f[d(q, x k )] = 6'[f (q), x k ] =s> #(C x [f (q)]) < n = #[C x (q)]. Q.E.Do 5 06c Corollary : Let h € G(A)o Then, for any q € Q and for any x € I, #[C x (q)] = #(C x [h(q)]). Proof : The corollary is immediate from # 5*5 (i) and the fact that h is monic o Q.E.D. 5.7° Definition : Let q € Q and x € I, where A = (Q, I, 6). C x (q) is said to be circular if, and only if, , #[C x (q)] 6(q, * ) = q. - 43 - In the example following definition # 5„3, C (p), C y (q), C (r), C x (q), and C x (r) are circular; but C x (p) is not, since 6(p, x- 5 ) = q ^ p. 5 o 8 o Lemma ; Let f be an endomorphism on A = (Q,I,6), and suppose for some q € Q and x € I, f(q) = 6(q, x) Then, f n (q) ■ 6(q, x n ), for all positive integers n„ Proof : The lemma clearly holds for n = 1„ Suppose k k that f (q) ■ 6(q, x ) , for some positive integer k Q Then, f k+1 (q) - f[f k (q)] - f[6(q, X k )] = 6[f(q), x k ] = 6[6(q, x), x k ] = k k+1 6(q, xx ) = 6(q, x ) D Thus, the lemma follows by induction. Q.EoD. Definition ; Let f be an endomorphism on A = (Q, I, 6)o For each q € Q 9 the f-cycle of q, denoted by c f (q) 9 is defined by, c f (q) = {f (q)s n = 0, 1, 2, ...}, n . . def ,.,- . . -, where f (q) = q c The length of the f-cycle of q is #[c f (q)j We say that c f (q) is circular if, and only if, f f (q) = qo We note that, for any q € Q, #[c f (q)] > 0, just as, for any x € I, #[c (q)] > o Furthermore, under the conditions of # 5o8, c„(q) ■ C (q) Indeed, this is a restatement of # 5.8 in J. -A. terms of definition # 5<>9, and we shall use the two statements interchangeably - w- - 5.10. Lemma : #[c f (q)] = the least positive integer, n, for which there exists a non-negative integer k < n, such that f n (q) = f k (q)o Proof : The proof of the lemma is almost identical to that of # 5 A and, therefore, we omit it. Q.E.D. 5.11. Theorem : Let h € G(A). Then, c h (q) is circular for any state q of A. Proof : Suppose Cv^O is not circular, and let #[c h (q)] = n. Then, h n (q) = p 4 q. But, p £ c h( there exists a least positive integer k < n, such that h n (q) = p = h (q) => by # 5«10, h " (q) ^ h (q). But then, two distinct states, h (q) and h (q), have the same value, p, under h. Hence, h is not monic, contradicting the hypothesis* Thus, c h (q) is circular. Q.E.D. 5.12. Corollary : (i) Let A = (Q, I, 6), let h (iii) This is immediate from (ii) and # 5°8„ Q E.D 5°13q Lemma ; Let h € G[A(q)]o Then, h is completely determined by its value on q„ Proof ; Let p € 6 (q)o Then, there exists x € I*, such that p = 6(q, x) Since h is monic, we lose no generality by considering this representation of p, though others exist . Then, h(p) ■ h[6(qs, x)] = 6[h(q), x] and, hence, h(p) is determined by the value h(q) on q QoEoDc We shall use the above lemma # 5° 13 implicitly in the following discussion,, 5oi4o Theorem s Let q be a state of Ao A necessary and sufficient condition for the existence of a non-trivial auto- morphism on A(q) is the existence in A of a state p / q, - kG - which satisfies: (i) A(p) = A(q); (ii) C Y (q) is circular, where p = 6(q, x); (iii) 6(q, y) = 6(q, z) => 6(p, y) = 6(p, z), for any y and z in L Proof of necessity : Let h € G[A(q)] be non-trivial Then, by # ^„12 (or # 5-13), h(q) 4 q. Let h(q) = p = 6(q, x), since h € G[A(q)]. Then, q 4 h(q) € A(q) => there exists x € I, such that 6(q, x) = p. By # 5»2, A(p) = A[h(q)] = A(q) and, hence, p € gen A(q) and (i) is satisfied. By # 5»12, C (q) is circular and, thus, (ii) is satisfied. Let 6(q, y) = 6(q, z), for some y,z € I, Then, h[6(q, y)] = h[6(q, z)] => 6[h(q), y] = 6[h(q), z] => 6(p, y) ■ 6(p, z), and (iii) is satisfied. Proof of sufficiency : Let (i), (ii), and (iii) be satisfied and define a function h: A(q) — £> A(q) by, h[6(q, u)] = 6(p, u), for all u € I. Now, suppose 6(q, u) = 6(q, v), for some v € I. Then, by (iii), 6(p, u) = 6(p, v), so that h[6(q, u)] = h[6(q, v)] and, thus, h is well defined on A(q). Let #[C x (p)] = n. Since p € C x (q), C x (q) = C x (p)„ Then, since C x (q) is circular by (ii), h(q) = h[6(q, x 11 )] = 6(p, x ) = p 4 q, by # 5.12 (ill), and, hence, h is non-trivial. Let r = 6(q, u) be an arbitrary state of A(q). Then, - 47 - for any v € I, h[6(r, v)] = h(6[6(q, u), v] ) = h[fi(q, uv)] = 6(p, uv) = 6[6(p, u), v] = 6(h[6(q, u)], v) - 6[h(r), v]. Hence, h is an endomorphisnio Now, let r = 6(q, u) and s = 6(q, v) be two states of A(q), such that h(r) = h(s)o Then, h n (r) = h n (s) =-> h n [6(q, u)] = h n [6(q, ▼)] =-*> 6[h n (q), u] = 6[h n (q), v], since, if h is an endomorphism, so is ho But then, by # 5<>8, 6[6(q, x n ), u] ■ 6[6(q, x n ), v] => 6(q, u) = 6(q, v), since #[C (q)] = n Q Leo, r = s and, therefore, h is monic„ But, on a finite set, a monic endomorphism is epic and, thus, since Q 4 , . is finite, h is epic Hence, h is an automorphism. A(q) Q.E.D, The following examples satisfy all conditions of # 5»1^» save one, but not its conclusion., We note that, if C x (q) is circular, then 6(q, x) € gen A(q), so that an example separating these two conditions cannot be given, (a) Let A be the trivial automaton; i e , Q = {q} , M = {x} , and 6(q, x) = q 6(q, z) = 6(p, z), for all z € I => if 6(q, u) = 6(q, v), then 6(p, u) = 6(p, v), for any u,v € L Thus, all conditions of # 5°1^ are satisfied, except that C x (q) is not circular and, indeed, the only auto- morphism on A is the identity » (c) Let A = (Q, I, 6), with Q = {p, q, r, s, t}, M = {x, y, z}, 6(q, y) = 6(q, z) = 6(t, x) = 6(t, y) = 6(t, z) = t, 6(p, z) = 6(s, x) = 6(s, y) = 6(s, z) = s, 6(p, x) = q, 6(p, y) = 6(r, x) = 6(r, y) = 6(r, z) = r, and 6(q, x) = p„ y,z x,y,z x,y,z x,y,z Here, there exists p t q, such that A(p) = A(q), and C Y (q) is circular. However, condition (iii) is not satisfied and, again, the only automorphism on A is the identity. Theorem § 5.14 furnishes the basis for determining the existence and nature of the non-trivial automorphisms on an automaton, as will be seen below, when this theorem is applied - ^9 - to the primaries of the automaton, since an automorphism on A maps each primary onto a primary . Thus, with the aid of theorem # 5°1^ and the remarks on extensions of automorphisms in section # 7, below, all automorphisms on A which map each primary onto itself are accounted for and characterizedo The remaining auto- morphisms are those which map at least one primary onto another, and those exist only when two distinct primaries are isomorphic, — a fact which can be discovered by inspection, if the automata are not too large „ In theorem # 5°l^s a non-trivial automorphism h € G[A(q)J was shown to be characterized by the action of a member x of I on a designated, initial state q In theorem § 5«15s we show that such an automorphism partitions gen A(q) into disjoint cycles of equal lengths 5°15q T heorem; Let h € G[A(q)] be non-trivial, where h(q) - 6(q, x) D Suppose there exists r € gen A(q) - C (q)o Then (1) #[>y(r)] = #IX(q)] h' •hr — -x (ii) c, (r) n C (q) = Proof ? (i) Let #[C x (q)] = n and #[c h (r)] = m By ## 5o8 and 5»12, #[C x (q)] = n =£> h n is the identity on A(q) =^> m < n D But, by # 4„12, h m is the identity on A(r) = A(q) and, thus, by # 5o8, n = #[C_(q)] < m c Hence, n = m (ii) By # 5ol^ s there exists s € Q^( q ) such that h(r) - s 4 r, A(s) = A(r) = A(q), C (r) is circular, where - 50 - s = 6(r, y), and 6(r, z) = 6(r, t) => 6(s, z) = 6(s, t) for any z, t € I. Now, by # 5°8, c h (r) = C y (r), and it remains to show that C (r) (1 C x (q) = 4 . Suppose that p € C_(r) n C x (q)o Then, there exist smallest non-negative integers, i and j, such that 6(q, x ) = p = 6(r, y j ), But then, by # 5*8, h i (q) = h J (r)„ If j < i, then r = h°(r) - h i_,j (q) = 6(q, x 1 "" 3 ) => r € C x (q). If, on the other hand, i < j, where #[c h (r)] = n, then h (q) = h n (r) = r =s> r € G x (q)c Since both alternatives produce a contradiction to the hypothesis, c h (r) n C x (q) = <£•> Q.EcDo We denote the order of a group element, Y9 by o(y) 5cl6c Corollary ; Let h € G[A(q)], where h(q) = 6{q, x)o Then, o(h) = #[C x (q)] = #[c h (q)]. Proof : The corollary follows from ## 4d2 and 5° 15° QoEoD, Theorem # 5«17 is merely a collection of technical lemmas, presented here primarily to afford facility of manipu- lation. Yet, it displays such interrelationships as the following? c h (q x ) = C x (q 1 ) and c h (r 1 ) = CyCl^), for h € G[A(q)] and q l» r l ^ gen A ^5» then respective h-successors of q^ and r^ are connected in the same manner q-^ and r, are connected, as is illustrated in the following partial diagram of an automaton. 51 - I /" y U- 1 r 2 5ol7° Theorem s Let A(q) = A(r), let h € G[A(q)], let h(q) ■ p » 6(q s x) 9 and let h(r) = s = 6(r, y) = 6(q, z). Then, for all non-negative integers k, (i) 6(q 9 x k z) - 6(q 9 zy k ); (ii) r - 6(q 9 u) S b 6{p 9 U ); (iii) q « 6(r 9 v) p = 6(s 9 v). 6(r, y k ) = 6(q, x k u) (q, x k ) = 6(r 9 y k v) (ii) With k = 0, 6(r, y k ) = 6(q, x k u) Proof; (i) 6(q, x k z) = 6[6(q 9 x k ) 9 z] = 6[h k (q), z] = h k [6(q 9 z)] - h k (s) = h k [h(r)] - h[h k (r)] = h [ 6 ( r , y k )] = 6[h(r), y k ] = 6[6(q 9 z) 9 y k ] - 6(q 9 zy k )„ e> r = 6(q, u) Now 9 r 6(q 9 u) =^=> h(r) h[6(q 9 u)] — > h(r) d[h(q), u] => s b 6(p 9 u) Furthermore 9 s = 6(p 9 u) ==> h(r) ■ h[6{q 9 u)] => for all k £ 9 h k (r) - h k [6(q, u)] — ► 6(r, y k ) = 6[h k (q), u] = 6[6(q 9 x ) 9 u] = 6(q 9 x k u) The proof of (iii) is almost identical to that of (ii) and 9 thus 9 we omit it Q Q«EoD - 52 - 6 18„ Theorem ; Let h € G[A(q)], let h(q) = 6(q, x), and let p € Qwn) Then, (i) o(h) = #[C x (q)] divides #[gen A(q)]; (ii) #[c h (p)] divides #[C x (q)] c Proof ; (i) By # 5-16, o(h) = #[C x (q)] and by # 5.15, for any r € gen A(q), C x (q) and c n ( r ) have the same length and are either disjoint or identical . Hence, h partitions gen A(q) into disjoint cycles, each of length o(h) and, therefore, o(h) divides #[gen A(q)]o (ii) Let o(h) = n and #[c^(p)] = m c Clearly, m < n and, hence, n = sm + t, for some positive integer s and some n sTn*t*t non-negative integer t < m c Now, p = h (p) = h (p) = h [(h m ) S (p)] = h (p) => t = 0, since t < m ■ #[c h (p)]. But then, n = sm =^=o m divides n„ QoEcDo We note that the existence of a generator, r, of A(p), for which r = 6(p, y) and such that #[C y (p)] does not divide #[gen A(q)] y does not indicate that G[A(q)] is trivialo It merely shows that, for no h € G[A(p)] does h(p) = r As an example, let A = (Q, I, 6), with Q = {p, q, r, s}, M = {x, y} , &(p, x) = 6(s, x) - q, 6(p, y) = 6(s, y) = r, 6(q, x) = 6(r, x) = p, and 6(q, y) = 6(r, y) = s„ - 53 - |y y Here, #[C (p)] = 3, while #[gen A(p)] = 4. Yet, h € G[A(p)], where h(p) = q (and, hence, h(q) = p, h(r) = s, and h(s) = r). Indeed, h is the only non-trivial automorphism on A(p)o 5q19q Definition : The set of automorphic images of a state q of an automaton A, denoted by G(q), is defined by, G(q) = {h(q): h € G(A)} def For any subgroup H of G(A), H(q) = {h(q)s h € H} 5 o 2 „ Lemma ; Let A = (Q, I, 6) and let q € Q Then, (I) [G(q), #] is a group, where "# H is defined by, h(q) # g(q) = hg(q), for any h,g € G(A); (ii) [G(q), #] is a homomorphic image of G(A); (iii) If A = A(q), then [G(q), #] * G(A)„ Proofs Let 9? G(A) — > [G(q) 9 #] be defined by, 9(h) - h(q), for all h € G(A)o Then, 9 is well defined, since each h € G(A) is well defined on Q Let h,g € G(A) Then, 9(hg) = hg(q) = h(q) # g(q), and 9 is a homomorphism Thus, (ii) is proved and, since 9 is clearly epic, (i) is provedo To prove (iii), we note that q € gen A and, hence, by # iJ-oll, distinct members of G(A) have distinct values for q„ - 5^ - Hence, #[G(q)] = #[G(A)] and 9 is moniCo Thus, 9 is an isomorphism Q.EoDc 5o21o Lemma ; Let A be an arbitrary automaton ^ let H be a subgroup of G(A), and let p and q be any two states of Ao Then, either H(p) and H(q) are identical, or they are disjointc Proof ; Let t 6 H(p) n H(q)o Then, there exist h,g € H, such that t = h(p) = g(q) ssaa *> g~ h(p) = q c Now, for any k € H, k(q) = kg =1 h(p) € H(p) =^> H(q) £ H(p)» By the symmetry of the argument, we also have H(p) c H(q) and s hence, H(p) = H(q)o QoEoDo 5°22 u Theorem ; #(G[A(q)]) divides #[gen A(q)]„ Proof ; By # 5»21, G[A(q)] partitions gen A(q) into disjoint subsets of the form G(p) and, by # 5 o 20, for each such G(p), #[G(p)] = #(G[A(q)])„ Hence, #(G[A(q)]) divides #[gen A(q)]. QcEcD, We note that # 5,18 (i) follows from # 5„22 as a corollary , Fleck states as theorem 2 in [5]; If A = (Q, I, 6) is strongly connected automaton, then #[G(A)] < #(Q). Since a - 55 - strongly connected automaton is the minimal automaton of each of its states and, since in that case gen A = Q, Fleck's theorem is an immediate corollary to theorem # 5»22 In fact, Fleck's theorem can be strengthened into the following corollary, which follows immediately from theorem # 5°22 and, hence, is stated without proof o 5o23 3 Corollary ; Let A = (Q, I, 6) be a strongly connected automaton Then, #[G(A)] divides #(Q)» We recall at this point that two members, x and y, of I* are distinct for A = (Q, I, 6) if, and only if, there exists q € Q such that 6(q, x) 4 6(q v y) Thus, since Q is finite, there exists only a finite number of members of I* distinct for A (even though each may be represented in an infinite number of ways)o It is in this sense that we use the term "distinct" in # 5°24 and other results to follow. 5o2^o Characterization Theorem ; Let A(q) = (Q, I, 6), and let {x i ; 1 < i < n} be a maximal set of distinct members of I* satisfying; (i) C x (q) is circular; (ii) for any y,z € I*, 6(q, y) - 6(q, z) =£> 6(q 9 x i y) = 6(q, x^); (iii) 6(q, x 1 ) = 6(q, x ^ ) => i = j (1 < i, j < n). Then, (a) any h € G[A(q)] is defined by h(q) = 6(q, x ± ) , for some i € {1, 2, 00 o, n} ; - 56 - (b) "for all u € I*, h 1 [6(q, u)] = 6{q, Xju)" defines an automorphism h* on A(q) for each i € {1, 2, „ „ „ , n} ; (c) #(G[A(q)]) = n» Proof : (a) This is immediate from # 5d4 (b) For each i € {1, 2, a „ „ , n} , since C x (q) is circular, there exists a positive integer k, such that q = &(q, x i ) = 6[6(q, x i ), x, " ] and, hence, q is a state of A[6(q, x,)]o Thus, A[6(q, x^] - A(q)o Hence, unless 6(q, x i ) = 6(q, €) = q and h, is the identity automorphism, (b) follows from # 5cl4 u (c) From (a), we have #(G[A(q)]) < n and, from (b), we have n < #(G[A(q)]). Hence, #(G[A(q)]) = n, QoEoDo 6o PRIMARIES AND AUTOMORPHISMS 6ole Theorem : Let f: A = (Q, I, 6) — > A 1 = (Q*, I, 6') be a homomorphism. Then, for each primary P" of f(A), there exists a primary F of A, such that f(P) = P*. Proof : Without loss of generality, let f(A) = A'. Let p f € Q* be such that P* = A(p')<> Then, there exists p € Q, such that f(p) = p' Now, by # 5.1, f[A(p)] = A[f(p)] = P*. By # 2o5, there exists a primary A(r) of A, such that p € Q^( r ) and, hence, there exists x € I*, such that 6(r, x) = p„ Now, p« = f( p ) = f[6(r, x)] = 6»[f(r), x] and, hence, A(p*) « A[f(r)] But then, by # 2 A, A(p«) = A[f(r)] = f[A(r)] Since A(r) is a primary of A, the theorem is proved, Q.E.D. The reader will note that theorem #6.1 cannot be strengthened to read, "the homomorphic image of a primary is a primary of the homomorphic image", as may be seen from the follow- ing example: Let A = (Q, I, 6), with Q = {p, q, r, s}, M = {x} , 6(p $ x) = q, and 6(q, x) = 6(r, x) = 6(s, x) = r Let f be defined by, f(p) = p, f(q) = f(s) = q, and f(r) = r. Then, f is a homomorphism on A D Now, P= ({s, r}, I, 6) isa - 57 - - 58 - primary of A, yet f(P) = A(q) is not a primary of f(A) = A(p). As might be expected, the stronger statement may be made when f is an isomorphism, as is illustrated by the following theorem 6o2, Theorem ; Let B be a primary of A = (Q, I, 6) and let h: A — ■ > A' = (Q'» I, 6 9 ) be an isomorphism. Then, h(B) is a primary of h(A) Proof : Since B is a primary of A, there exists q € Q B , such that B ■ A(q) Then, by # 5.1, h(B) = A[h(q)]. Suppose A[h(q)] is not a primary of h(A)„ Then, there exist p* € h(Q) - h(Q B ) and x € I, such that 6(p s , x) ■ h(q) Q But, since h is an isomorphism, h(Q) - h(Q B ) = h(Q - Q„) and, hence, there exists p € Q - Q B such that h(p) = p 8 Now, h(q) = 6«(p\ x) = 6°[h(p), x] = h[6(p, x)] and, since h is monic, q = 6(p, x). But, p i Q A ( q ) and q = 6(p, x) => A(q) 5 A(p) ==i> A(q) = B is not a primary of A, contradicting the hypothesise Hence, h(B) is a primary of h(A) u QoE.Do 6,3q Corollary ; An automorphism of an automaton A maps each primary of A onto a primary of A. Proof: The corollary is immediate from # 6„2 QoE.D. 6.4, Definition : An automorphism h € G(A) is said to se P-rep;ular If, and only if, h(B) = B, for each primary B of The set of P-regular automorphisms on A is denoted by P(A). - 59 - 6o5q Theorem ; For any automaton A, P(A) is a normal subgroup of G(A)» Proof ; Let f,g € P(A) e Then, for any primary B of A, f(B) - B - g"" 1 (B) =*=> fg" 1 (B) = fCg^CB'J] = f(B) = B =*> P(A) is a subgroup of G(A)o Let h € G(A), let k € P(A), and let A(q) be a primary of Ao Then, by ## 5°1 and 6o3, h[A(q)] = A[h(q)] is a primary of A => k(A[h(q)]) = A[h(q)] ==o h~ 1 kh[A(q)] = h^hlXq)]) = A(q) Hence, h -1 kh € P(A) and, therefore, P(A) is a normal subgroup of G(A)° Q.E.Do As the following theorem is immediate from ## 5»1^ and 6o3» we state it without proof 6060 Theorem ; Suppose #(gen B) = 1, for each primary B of an automaton Ao Then, (i) P(A) is trivial; (ii) If G(A) is not trivial, then each non-trivial member of G(A) maps at least one primary of A onto another; (iii) If 9 in addition, no two primaries of A have the same number of states, then G(A) is trivial. 6o7o Theorem ; Suppose that, for every pair of primaries, B and C (B 4 C), of A, at least one of the following holds; (i) #(gen B) 4 #(gen C); (ii) #(Q B ) 4 #(Q C ) - 60 - Then, P(A) = G(A) D Proof ; Let B = A(q) be a primary of A, and let h € G(A)o Then, by # 5.1. h(B) = A[h(q)]; I.e., the image of a generator of B is a generator of the image h(B) of B. But, by # 6o3, h(B) is a primary of A and, hence, h maps gen A(q) onto gen h(B). Thus, since h is monic, #(gen B) 4 #(gen C) =>h(B) 4 Co If, on the other hand, #(Q B ) 4 #(Q C ), then h(B) ^ C, by # 6„3, since h is monic. Hence, in either case, h(B) = Be But, since B was arbitrarily chosen, G(A) = P(A). Q.E.D. 6080 Theorem ; Let B-j_, B 2 9 • ° • » B n be all the (distinct) primaries of an automaton A, and let q, € gen B*, 1 < 1 < n e Let h € P(A), and let h(q 1 ) = 6(q., x. ) , 1 < i < n Then, o(h) = lcC u m {#[C x (q 1 )]: 1 < i < n} . Proof; Let h j _ be the restriction of h to B 1 , let e i denote the identity on B,, let k^ denote the order o(hj_) of h^ in G(Bj), and let l.c.m.{k,: 1 < i < n} = m Q Then, for each i € {1, 2, . „ , n} , h 1 = e^ But then, h m is the identity on A and, hence, o(h) < m Q Let p be a positive integer, such that p < m Then, there exists i € {1, 2, 000 , n} , such that k t does not divide p. Hence, there exist non-negative integers, q and r, such that p = ql^ + r, where < r < kj_ „ Hence, h p = h^o But, since < r < k 1 and 0(1^) = k^ h^ ¥■ e^. Hence, h t p 4 e 1 ,P and, therefore, h p 4 e (where e is the identity automorphi sm - 61 - on A) Thus, o(h) 4 P, for all positive integers p < m. Hence, o(h) = m Now, by # 5»l6, k i = #[c x (q^], for each i 6 {1, 2, . . e , n} , and, by # 5«15» it is immaterial which generator of a primary is selected. The theorem follows Q.E.D. 7° EXTENSIONS OF AUTOMORPHISMS In sections ## 5 and 6, the automorphisms of automata of rank 1 were characterized and counted. Consequently, we have a considerable amount of information about the automorphisms of each primary of an automaton „ However, if two primaries have a non-empty intersection, an automorphism of one primary may con- flict with an automorphism of the other on the common domain* In this section, we approach the study of the P-regular automorphisms of an automaton, whose rank is greater than 1, by starting with an automorphism of one primary and studying its extensions to other primaries . First, however, we reduce the problem to its essential scope 7.1. Definition ; Let F be a subset of the set of primaries of an automaton A. Then, U P is said to be a P€H» component of A if, and only if: (i) for any two primaries, B and C, in E>, there exists a sequence B = P-p P 2 , . „ . , P r = C of primaries in ff, such that P t n P 1+1 ^ A(, then D fl P = A((zO, for all P € S Thus, a component is a minimal separated subautomaton The following lemma # 7 °2 is immediate from the definition # 7„1 and, thus, we state it without proof. - 62 - - 63 - 7 c 2 , Lemma ; (i) Each primary of an automaton A is contained in a unique component of A» (ii) Every automaton is uniquely decomposable into components o (iii) Any two components of an automaton are either disjoint or identicalo (iv) Any union of components of an automaton A is a separated subautomaton of A„ 7o3q Theorem ; Let h: A = (Q, I, 6) — * A be an iso- morphism, and let C be a component of Ao Then, h(C) is a component of h(A)« Proof; By # 6 C 2, if B is a primary of A, then h(B) is a primary of h(A)o Now, let B]_, B 2 £= Q 9 such that B]_ P B 2 ¥■ do Then, clearly, hCB-^) n h(B 2 ) 4 6* Hence, if P-^, P 2 , . .., P r is a sequence of non-dis joint primaries of A, then, h(P]_)j h(P 2 ), ooo, h(P r ) is a sequence of non-dis joint primaries of h(A)o Thus, the primaries of h(C) satisfy (i) of # 7 1 Now, let D 9 be a primary of h(A), such that D 9 n h(P) ^ A(gO, for some primary P in Ao Since h is an iso- morphism, so is h and, hence, h (D* ) is a primary of A, by # 6 2 Let h" 1 ^ 1 ) = Do Then, h(D) n h(P) 4 A(*0 =s> D P P ¥ since h is moniCo Hence, if P is one of the primaries in C, so is Do Thus, if C = JJ P«, then D 9 = h(F i ), for some i € {1, 2, oo„, k} => D 9 « h(C)o Hence, h(C) is a component of h(A) QoEoD, - 64 - 7o4o Corollary ; Let h € G(A)„ Then, h maps each component of A onto a component of Ac Proof ; The corollary is immediate from ## 7.2 and 7.3. QcEoD. In contrast to # 7o3» the homomorphic image of a component need not be a component of the homomorphic Imager as can be seen from the example in remark # 4 15° 7°5q Definition ; An automorphism h € G(A) is said to De C-regular if, and only if, h(C) = C for each component C of Ac. The set of all C-regular automorphisms on A is denoted by C(A) 7 060 Theorem ; For any automaton A, P(A) is a normal subgroup of C(A) and C(A) is a normal subgroup of G(A) Proof ; Let f,g 6 C(A)o Then 9 for any component C of A, f(C) = C = g-^C), by # 7o4„ Hence, fg = ' 1 (C) = f[g" =1 (C)] = f(C) = C, and thus, fg" 1 € C(A) Hence 9 C(A) is a subgroup of G(A). Let h € G(A), let k € C(A), and let C be a component of A, Then, by # 7.4, h(C) is a component of A => k[h(C)] = h(C) ==> h _1 kh(C) = h~ 1 [h(C)] = C„ Hence, h^kh € C(A) and, therefore, C(A) is a normal subgroup of G(A). Now, if h € P(A), then h(P) = P, for all primaries P A => h(C) = C, for all components C of A ==> h € C(A) => - 65 - P(A) £ c(A)„ But then, by # 6 5, P(A) is a normal subgroup of C(A) Q.E.D, For ease of reference, we gather some obvious, but useful, facts in a single theorem., 7»7. Theorem ; Let n be the number of distinct com- ponents of an automaton A, and let r(C) denote the number of distinct primaries of a component C of A. Then, (i) If n = 1, then G(A) = C(A). (ii) If r(C) = 1 for all components C of A, then P(A) = C(A)o (ili) If C(A) is trivial, then any non-trivial auto- morphism of A maps at least one component of A onto another. (iv) If r(C) 4 r(D), for every pair, C and D, of components of A, then G(A) = C(A). The proofs of these results are all immediate. 7°8° Definition ; Let U and V be subautomata of A. An automorphism h € G(U) is said to be extendable to V if, and only if, there exists an automorphism h 9 € G(V), which is identical to h on the common domain U n V. The extension of h to V by h' is the function h v h»; U U V — i > U U V, defined by, (h(q), if q € Qrr, (h v h«)(q) = { {h«(q), if q € Q V o - 66 - We say that h v h 9 is an extension of h (alto, h') to V (alto , U) o It is immediate from the definition #7-8 that, h v h* € G(U U V) and that, if U and V are primaries of A, then h v h' € P(U U V)» We shall make implicit use of this fact, 7 „ 9 o Lemma ; Let U, V, and W be subautomata of Ac Let h € G(U), let h v k be the extension of h to V by k € G(V), and let (h v k) v j be the extension of h v k to W by j € G(W) Then, (i) h v (k v j) is defined and is the extension of h to V U W by k v j; (il) [(h v k) v j](q) = [h v (k v j)](q) 9 for all q € Q U U V U W" Proof: (i) Let q € Q y p r Then, q € Q (y y y) p y and, hence, j(q) = (h v k)(q),. But, q € ^ => (h v k)(q) = k(q) = - J(q) = k(q) u Hence, k v j is defined and is the extension of k to W by jo Now, let p(Q un(vuy) » If p € Q y n v , then h(p) = k(p) by hypothesis,, But, since k v j is an extension k and p j(p) = - 67 - (k v j)(p) =s> h(p) = (k v j)(p). Hence, h and k v j are identical on their common domain. Since kvj€G(VUW), h v (k v j) is the extension of h to V U W by k v j . (ii) If q € Qy, then (h v k)(q) = h(q) => [(h v k) v j](q) = (h v k)(q) = h(q), by definition # ?.8. Also by # 7 8, [h v (k v J)](q) = h(q). If q € Q v , then (h v k)(q) = k(q) ==> [(h v k) v j](q) = k(q), by definition # 7„8„ Also by # 7.8, (k v j)(q) = k(q) =s> [h v (k v j)](q) = k(q). If q € Q w , then [(h v k) v j](q) = j(q), by definition # 7<»8. Also by # 7o8, (k v j)(q) = j(q) ==> [h v (k v j)](q) = (k v j)(q) = j(q) Thus, for all q € ^ y y (J tf , [ (h v k) v j](q) = [h v (k v J)](q). Q.E.D. 7 o 10o Theorem : If U is a primary of an automaton A and h € G(U), then h is extendable to A if, and only if, h is extendable to the component of A which contains U. Proof ; Let C be the component of A which contains U, and suppose h is extendable to C« Let h v h' be the extension of h to C by h 9 . By ## 7.1 and 7o2, A - C « A (where A - C is the automaton A(Q - Q c ))» since both C and A - C are separated. Let e denote the identity automorphism def on A - C and define k = (h v h') v e. Clearly, k € G(A) and, since A - C is separated, k is the extension of h v h* to A (we allow ourselves this liberty with the language of definition § 7.8, as no ambiguity arises) by e, by # 7»3o But - 68 - then, by # 7c9, (h v h fi ) v e = h v (h° v e) Is the extension of h to A by h s v e„ Hence, h is extendable to A, Conversely, suppose that h is extendable to A., Let h v j be the extension of h to A by j and let J 1 be the restriction of j to Co Then, clearly, h v j* is the extension of h to C by j° and, hence, h is extendable to C Q.EoD, In view of theorem # 7„10, in studying extensions of automorphisms of primaries, we need concern ourselves only with one-component automata; i u e , connected automata „ In the remainder of this section, we assume each automaton to be connected u The study of non~C= regular automorphisms is comparable to that of isomorphisms between automata, on which considerable work has been done (see, for example, [3]K and which is outside the scope of this worko 7ollo Theorem ; Let U be a primary of an automaton A, and let h € G(U). If h is extendable to A, then h is ex- tendable to A[s(U)]c The converse does not hold* Proof : Clearly, if k is any extension of h to A, the restriction of k to A[s(U)] is an extension of h to A[s(U)] To show that the converse does not hold, we consider she following counterexample: Let A = (Q, I, 6), with Q = {q 1 l 1 < 1 1 9), M = {x, y, z) , 6(q x , x) = q 2 , 6(q £ , x) = q^ 69 6(q 3 , x) = 6(q 3> y) = 6(q 3 » z) = 6(q 2 , y) = 6(q £ , z) = 6(q 6 , y) = q^, 6(q^, x) = 6(q^, y) = 6(q^, z) = 6(q 19 y) = 6(q lt z) = 6(q^, y) = q 3> 6(q^, z) = 6(q g , x) = 6(q g , y) = 6(q g , z) = 6(q 9 , x) = 6(q 9 , y) = 6 (q , z) = q ?9 6(q^, x) = q 6 , 6(q 6 , x) = q^, and 6(q 6$ z) = 6(qp, x) = 6(q„, y) = 6(q^, z) = q go ^K x -^-«J) x,y,z The primaries of A are: B with Q B = {q-,, q 2 , q^» q/J » C with Q c = {q , q^, q^, q^, q , q g } , and D with Q Q = {q ? , q g , q^}. Define h on B by, h(q 1 ) = q 2 , h(q 2 ) = q 19 h(q ) = q^, and h(q^) = q 3 „ Then, h € G(B)o As is proved below, in # 7d7, the only extension of h to A[s(B)] =BUC is h v h 9 , where h9 (q 3 ) = q^i h"(q^) = q 3 9 h 9 (q 5 ) = q 6 , h 9 (q 6 ) = q^, h°(q ? ) = q 8 , and h e (q g ) = q„ Thus, h is extendable to A[s(B)] and it has a unique extension there . Now, since #[gen D] = 1 9 #[G(D)] - 1 by # 5 22 and, hence, the only automorphism on D is the identity „ But then, since (h v h 9 )(q,_) = q g 4 q 7 * h is not extendable to A, by QoEoDo We have stated the first part of theorem #7.11, in spite of the fact that it is immediate from previous results, - 70 - since it points the way to establishing the extendability of an automorphism of a primary to the entire automaton, A, and to obtaining all the P-regular automorphisms of As We start with an arbitrary primary, B, of A and list all non-trivial members of G(B) (These were characterized in ## 5ol4 and 5.24. The author has constructed algorithms, not included in this paper, for finding (i) the primaries of an automaton, (ii) the set of generators of a primary, and (iii) the non-trivial automorphisms of a minimal automaton and, hence, of a primary, the basis for the latter being theorems ## 5*1^ and 5°24o These algorithms are suitable both for hand computation and for computer application and are based on the transition table of the automaton) a We find the primaries, B-. , , B ?1 , 000 , B p ]_, other than B, which intersect s(B) (ioe , the primaries, other than B, in A[s(B)]) and use our results (primarily # 7ol7), below, to find and list all ex- tensions of each automorphism in the list of G(B) to B.,, In the same manner, we list all extensions of the automorphisms in the list of P(B U B^) to B 21 , and then to B-j, oo. s B r -j_ „ The end result is a list of all members of P(A[s(B)]). We now repeat the process; i e , we find the primaries, B 12» B 22» • 0, B r?2» in A[s(A[s(B)])] and not in A[s(B)], and the step-wise extensions in the above manner . The process ends when A is exhausted, or when a list of extensions is empty In either case, the list includes precisely the non-trivial auto- morphisms In P(A), where P(A) is trivial in the latter caseo That A must be exhausted if P(A) is not trivial, follows from fact that A has one component; i„e„, is connected., - 71 - It remains , then, to establish necessary and sufficient conditions for the existence of an extension of an automorphism of a primary to another primary and to count all such extensions. We devote the remainder of this section to this tasko The similarity of conditions in the following results to some of those previously encountered should not surprise the reader, as an extension of an automorphism is still an auto- morphism in itself „ The added requirement we have to satisfy here is that of finding an automorphism which agrees with a given automorphism on their common domain ; I^o, which maps specified states in the primary to predetermined states in the same primary „ In fact, we first deal with even more simplified version of the problem, — finding an automorphism which maps one specified state in the primary to another (possibly the same) specified state in the primary » We then study extensions of given auto- morphisms and, lastly, all P~regular automorphisms on the union of two primaries o 7ol2o Lemma; Let a,b € Q A (q)- If h € G[A(q)J is such that h(a) - b 9 then for each p € gen A(q), there exists r € gen A(q), such that: (i) C (p) is circular* where r = 6(p, x); (ii) there exists y € I*, such that a = 6(p, y) and b - 6(r, y); (ill) for any z,w € I*, 6(p, z) = 6(p, w) ^=s» 6(r, z) = 6(r, w) - 72 - Proof ; Let p 6 gen A(q). Then, there exists y € I*, such that a = 6(p, y). Let h(p) = r Then, by § 5„1, r € gen A(q); i e , the homomorphic image of a generator is a generator of the homomorphic image Now, b = h(a) = h[a(p, y)] = 6[h(p), y] = 6(r, y), and condition (ii) is satisf iedc . If p = r, conditions (i) (with x = g) and (iii) are trivially satisfied, and if p ^ r, conditions (i) and (iii) follow from # 5«1^ (the existence of some x € I* such that 6(p, x) = r is assured by the fact that p € gen A(q) Q Moreover, condition (iii) makes the choice of x immaterial ) Q QcE.Do 7° 13° Lemma ; Let a,b € ^A(q) and let p » r 6 gen A(q) be such that (i), (ii), and (iii) of lemma # 7.12 are satisf ied„ Then, h[6(p, u)] = 6(r, u), for all u € I», defines an automorphism h of A(q), such that h(a) = b» Proof : If p = r, then h is the identity on A(q), by # ^.12, and if p 4 r, then h € G[A(q)], by the proof of # 5ol4 or by # 5„24 (b); for, if r satisfies conditions (i) and (ill) of our lemma for p, then x satisfies conditions (i) and (ii) of theorem # 5.24, since r = 6(p, x) . Now, by (ii) of our lemma, a = 6(p, y) =j> h(a) = h[6(p, y)] = 6(r, y) = b c QoE.Do 7-14. Theorem : Let U and V be primaries of A, and Then, h is extendable to V if, and only if, 73 for some p € gen V, there exists r € gen V, such that: (i) C x (p) is circular, where r = 6(p, x); (ii) for each a € Qy « y, and for each y € I* such that a = 6(p, y), h(a) = 6(r, y); (iii) for any z,w € I*, 6(p, z) = 6(p, w) =s> 6(r, z) = 6(r, w)« Proof : Suppose that, for some p € gen V, there exists r € gen V, such that (i), (ii), and (iii) hold* Then, by # 7.13, h*[6(p, u)] = 6(r, u), for all u € I*, defines an automorphism h* of V, such that h-(a) = h(a), for each a ^ Qu n V° Hence, h is extendable to Vo Conversely, let h v h 9 be an extension of h to V by h* € G(V) Then, for each a € Q y n v , h s (a) = h(a) and, hence, by # 7ol2, for each p € gen V, there exists r € gen V, such that (i), (ii), and (iii) are satisfied,, Q.E.D. 7o15q Theorem : Let A(q) = (Q, I, 6), let 6(q, w) = r, and let {x^ : 1 < i < n} be a maximal set of members of I*, distinct for A(q) and satisfying: (i) C Y (q) is circular; *i (ii) for any y,z € I*, 6(q, y) = 6(q, z) => 6(q, Xjy) = 6(q, x ± z) ; (iii) 6(q, Xi ) = 6(q, x j ) => i = J (1 < i, J < n); (iv) if 6(q, x^w) = s for some 1 € {1, 2, O o , n} , then 6(q, x.w) - s for all j € {1, 2, O e , n) „ - ?4 - Then, (a) any h € G[A(q)] satisfying h(r) = s is defined by h(q) = 6(q, x^ ) , for some i € {1, 2, ..., n} ; (b) for each i € {1, 2, . .., n} , "for all u € I*, h i [6(q, u)] = 6(q, x^)" defines an automorphism h 1 on A(q) such that h^r) = s. (c) there exist exactly n distinct automorphisms h on A(q) satisfying h(r) = s. Proof ; Let h € G[A(q)] be such that h(r) = s. By # 5.2k, h is defined by h(q) = 6(q, x), where x € X and X is a maximal set of members of I* distinct for A(q) and satisfying conditions (i) 9 (ii)» and (iii) Since h(r) = s, we have s = h(r) = h[6(q, w)] = 6[h(q), w] = 6(q, xw ) and, hence, x satisfies all conditions required of the x* *s u But then, by maximality, for some i € {1, 2, . „ <, , n} , 6(q, x) = 6(q, x i ) and, hence, (a) is proved. From # 5.24 and the fact that each x 1 satisfies conditions (i) and (li), it follows that h^_ € G[A(q)] for each i € {1, 2, ..., n} . Since each x, satisfies condition (iv), s = 6(q, x^w) = 6[h 1 (q), w] = h^dCq, w)] = h^r). Thus, (b) is proved. Since the x^s satisfy condition (iii), (c) follows from (a) and (b). Q.E.D. - 75 - 7°l6o Lemma ; Let A(p) and A(q) be primaries of A, let h 6 G[A(p)], and let h(p) = 6(p, y), for some y € I*. A necessary and sufficient condition for the existence of an extension of h to A(q) is the existence of x € I* which satisfies: (i) C x (q) is circular; (ii) for any z,w € I*, 6(q, z) = 6(q, w) =£> 6(q, xz) = 6(q, xw); (iii) for each r € Q A ( p ) n A ( q ) and for any u,v € I*, 6(p, u) = r = 6(q, v) => 6(p, yu) = 6(q, xv). Proof : Let x € I* satisfy (i) and (ii) D Then, by # 5o2k, k(q) = 6(q, x) defines an automorphism on A(q)» Let r € Q A ( p ) n A ( q ) and let 6(p, u) = r = 6(q, v). Then, h(r) = h[6(p, u)] = 6[h(p), u] = 6(p, yu) and, similarly, k(r) = k[6(q, v)] = 6[k(q), v] = 6(q, xv). Now, if x satisfies condition (iii) as well, then 6(p, yu) = 6(q, xv) and, hence, h(r) = k(r)o Thus, h v k is the extension of h to A(q) by ko Conversely, suppose there exists an extension, h v j, of h to A(q) by j € G[A(q)], and let r = 6(p, u) ■ 6(q, v) Then, by # 7.15, there exists x € I* which satisfies (i) and (ii) and, in addition, j(r) = 6(q, xv) But h(r) = 6(p, yu) and h(r) = j(r) => 6(p, yu) = 6(q, xv) and, thus, (iii) is satisfied„ Q.E.Do - 76 - 7.17. Characterization Theorem ; Let A(p) and A(q) be primaries of an automaton A, let h € G[A(p)] and let {x : 1 < i < n} be a maximal set of members of I* distinct for A and satisfying: (i) C Y (q) is circular; x i (ii) for any z,w 6 I*, 6(q, z) = 6(q, w) =s> 6(q, x^) = 6(q, Xjw); (iii) 6(q, x ± ) = 6(q, Xj ) =*> i » J (1 < i, J < n); (iv) for each r € Q A / p \ n A(q)' wnere r = 6 ^ q » w * for some w € I*, h(r) = 6(q, x^w). Then, (a) any extension of h to A(q) is of the form h v k 1 , where ^ is defined by, k i (q) = 6(q, x i ) for some i € {1, 2, ..., n}; (b) for each i € {1, 2, „.., n} , h v k, defines an extension of h to A(q), where k, is defined on A(q) by, ^[atq, u)] = 6(q, x^u) for all u € I*; (c) there exist exactly n distinct extensions of h to A(q). Proof ; (a) and (b) follow from # 7.16 and condition (iv). Part (c) follows from (a) and (b) and condition (ill). Q.E.D. As a result of theorem # 7.17, we are able to arrive at a characterization and count of all members of P(B U C), where B and C are primaries of a given automaton, as is done in the following corollary # 7.18. - 77 - 7.18. Corollary ; Let A(p) and A(q) be primaries of an automaton A» Let it-,, t£» .«., t n ) be a maximal set of members of I* distinct for A and satisfying: (i) C t (p) is circular; (ii) for any z,w € I*, 6(p, z) = 6(p, w) => 6(p, ^z) = 6(p, tjw); (iii) 6(p, t ± ) = 6(p, t j) ==> i = j (1 < i, 3 < n). For each i € {1, 2, ..., n} , let {xj, , x^ , . . . , x i } be a maximal set of members of I* distinct for A and satisfying: (iv) C x (q) is circular; ^-k (v) for any z,w € I*, 6(q, z) = 6(q, w) =$> 6(q, x i k 2 ) = 6 ( ( 1» x i k w )j (vi) 6(q, x lk ) = 6(q, x ± . ) ==> k = j (1 < k, J < i^ ) ; (vii) for each r € Q A ( p ) n A(q)> 6(p ' fc i y) = 6(q ' x i k U ^ where r = 6(p, y) = 6(q, u) for some y,u € I*. Then, (a) any f € P [A(p) U A(q)] is of the form h, v g 1 , XL where h i is defined by h^p) = 6(p, t^), 1 < i < n, on A(p) and, for each such i, g* is defined on A(q) by g* (q) = k x k 6(q, x ik ), 1 < k < r l0 (b) For each i € {1, 2, „ . . , n} and for each k € {1, 2, „ o , r^}, f = hj_ v gj_ defines an automorphism f € ML P[A(p) U A(q)], where ^[Xp, u)] = 6(p, t i u) for all u € I*, and Si k [6(q, v)] = 6(q, x ik v) for all v € I*c n (c) #(P[A(p) U A(q)]) = 1 T ± . i=l - 78 - Proof ; The corollary is merely a consolidation of the two characterization theorems ## 5.24 and 7.17 (and the results preceding the latter), and it follows with no difficulty from these theorems. Hence, we spare the reader the repetitive nature of the detailed proof. Q.E.D. We close this section with a simple example, in which the reader will find an illustration to the results from #7.12 through # 7.18. Let A be the automaton having the following state diagram: x,y,z x,y,z x,y,z x,y,z We recognize in A three primaries: B with states p, q, r, s, a, b, c, d; C with states t, c, d; and D with states m, n, a, b. G(B) = {h, h 2 , h 3 , h 4 = e_}, where h(p) d = f r. We note 'B' that h and h J map a to b, b to to d, and d to c, maps each of a, b, c, d to itself, as does the Identity h = e B of B. G(D) = {k, k 2 = e D ) , where k(m) d = f n. maps a to b and b to a, both h and h 3 are - 79 - 2 extendable to D by k<, Similarly, h is extendable to D by o k = e D . On the other hand, G(C) = {e_} , where e G is the identity on Co Hence, there exists no extension of either h 1 2 or h y to C, while h is extendable to G by e c „ Thus, 3 2 P(B U D) = {h v k, h v k, h v e D , e B v e Q } , 2 P(B U C) = {h v e G , e B v e c ) , and P(B U C U D) = {h 2 v e Q v e D , e B v e Q v e Q } . 8. ADDITIONAL RESULTS AND POSTSCRIPT In this section, few lines of further investigation are suggested. The author has achieved some results in one such line, — quotient automata, and several of these results are reported first. The common definition of an Isomorphism between automata requires the latter to have the same input semigroup. Thus, the states of the domain automaton are relabeled by the isomorphism, but the inputs are not„ Yet, two automata may be essentially identical, but for the "names" of their states and inputs. We call two such automata essentially isomorphic , abbreviated by ^-isomorphic " (for a precise definition, see Bavel and Muller [10], where a "generalized homomorphism" is defined), 8,1. Representation Theorem ; Let H = (X, • ) be a finite group of order n. Then, there exists an n-state automa- ton A H of rank 1, such that G(A H ) » H » I, where I is the input semigroup of A H . Moreover, A H is unique up to e-iso- morphlsm and is strongly connected. We call A^ of # 8.1 the automaton-representation of the group H. We note that A H is a reversible automaton (see [10]), since its input semigroup is a group. By way of an example, consider the dihedral group D^ of t , defined by, o? = b =1, bab = a" 1 . A simple compu- - 80 - 81 - tation shows A n to be (essentially uniquely) the following automaton: "o - 8.2. Definition : Let A be an automaton. Then, the automaton induced by A (or, the induced automaton of A) is A /.»; i.e., the automaton-representation of the group of auto- morphisms of A„ As an illustration, if A is the automaton described just before #8.8, below, then its induced automaton is A~ D 3 described just before # 8.2, above. 8.3. Definition : Let A = (Q, I, 6). A triple A' = (Q% I*, Q') is said to be a semiautomaton of A if, and only if, A* is an automaton, where Q 1 £ Q f I' is a subseml- group of I, and 6' is the restriction of 6 to Q* x I*. 8.4. Theorem : Let H be a subgroup of G[A(q)]. Then, A(q) has exactly #[gen A(q)]/#(H) disjoint semiautomata, whose states are generators of A(q), each e-isomorphic to A«. (10) This definition is due to Barnes ([l5])» who uses the term "subautomaton" . We reserve the latter term for its customary use (see # o ?)o - 82 - 8.5. Theorem ; Let H be a subgroup of G(A), and define y on A by: y(q) = H(q), for all q € Q (see definition # 5.19). Then, y is a homomorphism. 8.6. Definition : Where y is the homomorphism of # 8.5, y(A) is the quotient automaton of A by H , denoted A/H. One approach to quotient automata is that of Give'on in [9], where he replaces an entire subautomaton by the trivial automaton (single-state automaton) . Definition # 8.6 yields quite a different class of automata. A comprehensive study of quotient automata should probably include a combination of both notions, with definition # 8 6 possibly generalized for less restricted homomorphisms on A. 8.7. Remark : If A is an automaton of rank 1, and H is a subgroup of G(A), then the generators of A/H are precisely the semiautomata of # 8.4. If, in addition, A is strongly con- nected, then (trivially) the states of A/H are these semiautoma- ta. Let M(H) denote the class of all automata A i , whose automorphism-group contains, as a subgroup, an isomorphic image tt^ of a given group H. One may study the "essential" differ- ences among members of Pk(E) by comparing the quotient automata Aj/H^ The following subclasses of AA(H) may yield interesting ~vat Ions when treated thus: (a) automata of rank 1; - 83 - (b) strongly connected automata; (c) automata whose entire group of automorphisms is isomorphic to H; (d) the intersection of (a) and (c); (e) the intersection of (b) and (c). A special role is played by the normal subgroups of the automorphism-group of an automaton in connection with quotient automata , We illustrate this role by the following example. Let A be the automaton with the following state-diagram: x,y,z def def P Where h^) - q 2 and k(q 1 ) = q^, G(A) = {e, h, h , k, kh, kh 2 } = D 3 o p H is a normal subgroup of G(A), where H = {e, h, h }, and A/H has the following state-diagram: x,y,z - 84 - Here,G(A/H) = {e, k»}, where k^i^) = r v * ndeed > G(A/H) « G(A)/H. On the other hand, K Is not a normal subgroup of G(A), where K = {e, k} , and A/K has the following state-diagram: ^P8) )x ,y,z Here, G(A/K) is trivial and not the three-group, as might have been expected. 8.80 Theorem : Let H be a normal subgroup of G(A) Q (i) For each g € G(A), ft A — > A/H Is a homomorphism, g def where Yg(q) = Hg(q), for all q € Qo (il) For each g € G(A), g' is an automorphism on A/H, where g' is defined by, g'(p') = Hg(q) = {hg(q): h € H} , for each q € p* = H(p) € Q A / H - (ill) G(A)/H is isomorphic to a subgroup of G(A/H). As a consequence of theorem #8.8 and the remarks preceding it, it appears that an investigation of the normal sub- groups of G(A) and their particular role in quotient automata Is desirable. Such an investigation proves to be rather extensive and, at this point, we shall just mention two of the normal sub- groups of G(A), in addition to P(A) and C(A), already defined (in # fj.h, and # 7.5 respectively). - 85 - 8o9u Definition ; f € G(A) is s.c , -regular if, and only if, f (B) = B, for all strongly connected B « A„ The set of all SoCo -regular functions on A is denoted by F(A)o 80IO0 Definition ; k € G(A) is completely regular if, and only if, k(B) = B, for every B « A» The set of all com- pletely regular functions on A is denoted by K(A) U We note that P(A) and K(A) are normal subgroups of G(A) and that we may have, {e} 5 K(A) 5 F(A) 5 G(A)<, The author has partially studied a variety of normal subgroups of G(A), and has obtained several interesting results concerning them c However, considerably more should be looked into before this investigation may be organized; for example, composition series of automata and their correspondence to the generating series of subgroups, Jordan-Holder type of treatment, and the like One other interesting avenue of investigation is the generalization of the present work to monadic algebras » As an illustration, the definition of primary may not be useful or, Indeed, may not even apply to an algebra, as is the case with the algebra of all integers under addition of 1» To generalize to the algebra A = (Q, M, 6), we call R c Q singly generated if, and only if, there exists r € Q, such that R c 6(r) We - 86 - call H c Q finitely closed if, and only if, every finite subset of R is singly generated by a member of Re Then, we call (R, M, 6) a primary of A if, and only if, R is a maximal finitely closed subset of Q. An obvious direction is that of applying such tools as are presented in this paper to sequential machines with output. We expect the source and the primary to be useful, for instance, in Investigating experiments, length of experiments (see, for example, Ginsburg [3]) and total length of experiments (see, for example, Bavel [l6]). In fact, these two structures are basic enough to be of use in most areas of the theory of automata and sequential machines. Another interesting area of investigation is the appli- cation of the results on automorphisms, presented and hinted at here, to group theory. As a particular example, the automaton- representation of a group may furnish the group theorist with a means of observing his finite group "in action", and draw some additional insights as a result. Other avenues of mathematical investigation suggest themselves, as well as the possibility of finding interesting applications of these results in the various fields employing weighted, directed graphs (Hohn, Seshu, and Aufenkamp [l7])» i.e., automata, for the representation of the natural systems under Investigation. BIBLIOGRAPHY [I] Rabin, M Oo, and Scott , Do, "Finite Automata and Their Decision Problems o H IBM j * Res, Dev G 3 (April 1959) 11^ - 125 [2] Ginsburg, So, "Some Remarks on Abstract Machines " Trans. Amer u Mathc Soc D 96 (I960), 400 - Wk [3] Ginsburg, So, An Introduction to Mathematical Machine Theory , Addi son-Wesley Publishing Coo, Inc«, Reading, Massachusetts (1962)o [4] Fleck, A e Co, "Structure Preserving Properties of Certain Classes of Functions on Automata " Computer Laboratory, Micho Sto Uo, May 19 61 (unpublished paper) „ [5] Fleck, A Co, "Isomorphism Groups of Automata " Jo ACM 9»^ (October 1962) 469 - ^76o [6] Weeg, Go Po, "Some Group Theoretic Properties of Strongly Connected Automatao" Computer Laboratory, Mich„ Sto Uo, May I96I (unpublished paper) [7] Weeg, Go Po, "The Structure of an Automaton and Its Operation Preserving Transformation Group „" J ACM 9 9 3 (July 1962) 3^5 - 3^9 o [8] Moore, E, F B , "Gedanken Experiments on Sequential Machines . H In Automata Studies , Princeton Uo Press, Princeton, New Jersey (1956) 12 9 - 153 ■ [9] Give "on, Y , "Toward a Homological Algebra of Automata " Depto of Commo SCo, Uo of Michigan, February 1965 (unpublished paper) [10] Bavel, Zo, and Muller, D E., "Reversibility in Monadic Algebras and Automata u " Proc Sixth Annual Symposium, Switching Circuit Theory and Logical Design, 1965 (to be published) e [II] Huzino, So, "On Some Sequential Machines and Experiments " Mem Faco Scio Kyushu UniVo, ser A, 12 (1958), 136 - 158 [12] Huzino, So, "Theory of Finite Automata . * Menu Fac Scio Kyushu UniVo, ser„ A, 15 (196l), 97 - 159» [13] Gluschkow, Theorie der Abstrakten Automates Veb Deutscher Verlag der Wissenschaften, Berlin (1963)0 [14] Kelly, J Lo, General Topology, Do Van Nostrand Co Q , Inc., Princeton, New Jersey (1955)° 87 88 - [15] Barnes, Bo, "Relationships Between Groups of Isomorphisms and Set of Equivalence Classes of Input for Automata." (Presented to ACM National Conference at Denver, Colorado. August, 1963 . Unpublished paper ) [16] Bavel, Zo, "On the Total Length of an Experiment, I." Proc . Fifth Annual Symposium, Switching Circuit Theory and Logical Design, 1964„ [17] Hohn, P. E , Seshu, S., and Aufenkamp, D. D., H The Theory of Nets." I.R0E0 Transactions on Electronic Computers, Vol. EC - 6, September, 1957c # 3, 15^ - 161. /IT A. Zamir Bavel was born in Tel-Aviv, Israel, on February 8, 1929. He graduated from Levinsky Teachers College in Tel-Aviv in 19^7, and was a teacher and assistant principal in Tel-Aviv until 1952. In 1952, he entered Southern Illinois University at Carbondale, Illinois, from which institution he received the A.B., B.S., and B.Mus. degrees in 195^ with highest honors. In 1955* he received the M.A. degree from the same institution, and taught there until 1962. He has attended the University of Illinois since then. He is a member of Delta Rho, Kappa Delta Pi, the American Mathematical Society, the Mathematical Association of America, and the Association for Computing Machinery. - 89 - AVI