LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510.84 I£6r no. 188-199 cop.Su Digitized by the Internet Archive in 2013 http://archive.org/details/partiallyordered189niev Report No. 189 >./V Lr ft? PARTIALLY ORDERED CLASSES OF FINITE AUTOMATA 3 by Jurg Nievergelt September 30, 1965 Report No. 189 PARTIALLY ORDERED CLASSES OF FINITE AUTOMATA* by Jurg Nievergelt September 30, 19^5 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, August, 19&5 • ACKNOWLEDGMENT The author wishes to express his gratitude to Professor D. E. Muller for his guidance during the preparation of this thesis. The author is also indebted to the University of Illinois for a fellowship which made this work possible, and to the Department of Computer Science with whom it has been his privilege to be associated over the past three years . 111 TABLE OF CONTENTS Page 1. INTRODUCTION 1 2 . BACKGROUND, TERMINOLOGY AND NOTATION h 3. THE SET OF FINITE AUTOMATA ON A FIXED INPUT ALPHABET ... 7 k. CONGRUENCES ON AN AUTOMATON AND ON ITS SEMIGROUP 13 5. PROPERTIES PRESERVED UNDER HOMOMORPHISM 28 6. CERTAIN LATTICES OF AUTOMATA 33 7. THE PARTIAL ORDER DEFINED BY SUBAUTOMATA 38 8. CONCLUDING REMARKS k2 9. BIBLIOGRAPHY kh APPENDIX k$ VITA 59 Iv ABSTRACT PARTIALLY ORDERED CLASSES OF FINITE AUTOMATA Jurg Nievergelt Department of Mathematics University of Illinois, 19^5 Most early studies in the theory of sequential machines or finite automata have been concerned with a single automaton. In recent years, however, one has begun to turn attention towards the study of sets of automata, and the work on loop-free structures of sequential machines, initiated by Hartmanis and extended by Krohn and Rhodes among others, is a good example of this current trend. Problems like: "given a number of automata, describe the class of automata which can be simulated by interconnecting an arbitrary number of copies of the given automata in all possible ways in series and in parallel" clearly illustrate this point of view. These studies of loop-free connection of automata suggest the following generalization: consider some set Y 1 I of finite automata, and define certain operations on it, like series and parallel connection of two automata, and certain relations, like one automaton being realized by another, with various possible definitions of the word "realize" -- then study the algebraic structure of |7| with respect to the operations and relations defined. This thesis represents a small step towards this goal. The problem under consideration is the following: on the set Try (A) of all finite automata (modulo isomorphism) having a given input alphabet A, define a partial order "<", where M' < M means M' is a homomorphic image of M. Study the poset m((a) and certain of its subsets. The main tool for studying the structure of JY[ (A) is the semi- group of transformations of the set of states associated with every automaton. The lattice of all such semigroups (under homomorphism as a partial order) is closely related to the poset Y'l(A),, and many of our results state that some subset of !fY\ (A), finite or infinite, is a lattice. It is shown that another partial order "c." on fY[ (A) , where M 1 c M means M' is isomorphic to a subautomaton of M, yields completely analogous results . 1 . INTRODUCTION Most of the studies in automata or sequential machine theory have been concerned with a single automaton. Problems concerning the reduction or minimization of an automaton and the sets of tapes accepted by an automaton have long been of central importance in the field. More recently attention has been turned towards sets of automata, and in particular to series -parallel (or loop-free) interconnection of automata, The early work by Hartmanis [1,2] and others led to a theory of machine decomposition, which deals with the question whether and how a given automaton can be realized from smaller automata connected to each other in series and in parallel. Krohn and Rhodes' approach [3] goes one step further. In addition to the above mentioned problem, it is also concerned with what might be called the problem of machine composition: given a number of automata, describe the class of automata which can be simulated by interconnecting an arbitrary number of copies of the given automata in all possible ways in series and in parallel. These studies of loop-free connection of automata point to the following generalization: consider the set 771 of all finite automata, and define certain operations on 7fl , like series and parallel connection of two automata, reduction of one automaton, etc., and certain relations on 7M , like one automaton being realized by another, with various possible definitions of the word "realize". Then study the algebraic structure of r) ft\ with respect to the operations and relations defined. This thesis represents a small step towards this goal. The problem we have posed ourselves is the following: -1- -2- Consider the set lflf[(A) of all finite automata (modulo isomorphism) having a given input alphabet A, and one binary relation (a partial order) "<" on ^(A), where M' < M means M 1 is a homomorphic image of M. Study the partially ordered set (poset, for short) lff[(A) and certain of its subsets. One reason for choosing a relation on ?T\(A) instead of some operation(s) is that Krohn and Rhodes 1 theory [3] can be interpreted as an investigation of the set of all finite automata under the operations of series and parallel connection, while no comparable study concerning relations is known to the author. The reason for choosing "homomorphic image" rather than some other relation, is that homomorphic images, or rather the very closely related partitions with substitution property on the set of states of an automaton, play a crucial role in the series -parallel decomposition (in Hartmanis : sense [1]) of automata, hence one can hope to find interesting relationships between the partial order : 'homomorphic image" and the two operations "series-" and "parallel-connection" . However, in Sec. 7 we show that one can get completely analogous results if one replaces < by another partial order c on fjf\(A) , where M' c M means M : is isomorphic to a subautomaton of M. Our main tool for studying the structure of Tf[{k) is the semigroup transformations of the set of states associated with every automaton. The .ce of all such semigroups under homomorphism is very closely related to TH(A), and many of our results state that some subset of 7V\(A), ite or infinite, is a lattice. -3- Throughout the whole investigation we come upon certain special equivalence relations 5 on lattices or general posets P such that the partial order defined on the poset induces a partial order on the set P/6 of all equivalence classes. The problem arises to find conditions under which P/8 is a lattice. The Appendix deals with this question in much more detail than is required for the main development, so the reader interested in automata is advised to read the first few pages of the Appendix and then refer to it only where it is mentioned in the text. 2, BACKGROUND, TERMINOLOGY AND NOTATION The presentation assumes familiarity with a few "basic concepts and results concerning partially ordered sets, lattices, equivalence relations and congruence relations. For reference see, e.g., Birkhoff [k] . The paper is self contained as far as automata theory is concerned. If 5 is an equivalence relation on a set S, we write s J 5s to indicate that the two elements s', s e S are in relation 6. s|S denotes the equivalence class to which s belongs, S/6 the set of all equivalence classes. In the lattice of all equivalence relations on a set (or congruence relations on a set with operations), the partial order of refinement is denoted by <, the operations l.u.b. (least upper bound) and g.l.b. (greatest lower bound) "by +, • respectively. If (3 < 6 are equivalences on S, then 5 defines in a natural we/ an equivalence relation on s/|3, denoted by o/f3, which satisfies (S/(3)/( 6/6) - S/S. The greatest equivalence (which relates all elements of a set S) and the least one (which relates no two distinct elements) are denoted "by 1, respectively, or by 1 , if there might be any ambiguity concerning the set involved. M = (Q, A, f ) denotes both a particular automaton and the class of automata isomorphic to it (p. 7). 7ft (A) denotes the set of all classes isomorphic finite automata with a given input alphabet A. In Sec. 7 (and only there), it includes the class of the empty automaton (p. 38) • -5- A* denotes the free semigroup on A, under concatenation (p. 7). S(M) = A*/ m (M) is the semigroup of the automaton M (p. 8 ) } where m(M) is a congruence on A* determined by M ( p. 8 ). u is an equivalence on *flf\(A) which relates automata with the same semigroup (p. 9). ■k, p usually stand for congruence relations on an automaton (p. 13), A(M) denotes the lattice of all congruences on M (p. 13). T usually stands for a congruence on a semigroup S(M), Z(M) denotes the lattice of all congruences on S(M) ( p. IT). The subset Z (M) (p. 23) of Z(M) consists of all congruences on S(M) which are the associated congruence s(n) (p. 18) of some congruence n on M. a is an equivalence on A(M) (p. 26) which relates congruences jt, jt 1 on M iff \s(tt) = s(jt'). 7 is another equivalence on A(M), called isomorphism (p« 13)> which relates congruences jt, it 1 on M iff M/jt = M/jt' (p. 7). - The mapping s is similar to an inverse of s and defines a congruence s(t) on M for every congruence T on S(m) (p. 20). , T is a property of automata which involves semigroups (p. .28).. Certain other properties of automata are defined on page 33- All these concepts arise in connection with the partial order < on ^(A), defined by homomorphic images (p. 8). - For the partial order c on 7)\(A) defined by subautomata (Sec. 7) some of these concepts are replaced by others: the role of congruences on an automaton is taken over by closed subsets of the set of states of an automaton (p. 38). . M' is a subimage of M iff it is isomorphic to a subautomaton (p. 38) of M. -6- At some places reference is made to certain concepts introduced in the Appendix, which are listed below. Order preserving (OP), p. 4 5 Stair relation, p. 4 5 Directed downward (DD) or upward (DU), p. 53* 3. THE SET OF FINITE AUTOMATA ON A FIXED INPUT ALPHABET Definition : A finite automaton M = (Q, A, f ) consist of a finite set Q of "states", a finite set A of "inputs", and a "transition function" f : QxA->Q. Definition : The free semigroup A* consists of all finite words ("input .: • ', strings") over A, including the empty string e, under concatenation as a "binary associative operation. The transition function f of an automaton M can be extended to a function (which will again be denoted by f ) f : Q x A* -+ Q by the following recursive definition: f(q,e) = q for all q e Q, and f(q,ax) = f(f(q,a),x) for all q 6 Q, a e A, x e A*. Definition : Let M = (Q, A, f), M' = (Q',A, f) be automata on the same input alphabet A. A homomorphism h from M to M' is a function h: Q - Q* such that h(f(q,a)) » f»(h(q),a) for all q e Q, a £ A. An isomorphism is a homomorphism which is 1:1 onto. If h is a homomorphism M - M 1 , then it is easily seen from the definition of extension of the transition function that h(f(q,x)) = f'(h(q),x) for all x e A*, q e Q. -7- -8- Definition : M' < M, or M' is a homomorphic image of M, iff there exists a homomorphism h: M -» M' which is onto. M 1 = M, or M 1 and M are isomorphic, iff there exists an isomorphism h: M -» M' . If h: M -> M 1 is an isomorphism, then h : M' -» M is also, hence M' = M implies M ! < M and M < M 1 . Conversely, if M' < M and M < M', M and M' must have the same number of states, and hence any homomorphism from M onto M' is 1:1, hence an isomorphism. Hence M' = M 4=> M' < M and M < M f . We will usually not distinguish between isomorphic automata, and by M we will denote both a class of isomorphic automata and any automaton which is a representative of this class. Let Hf\^A) be the set of all classes of isomorphic finite automata on the input alphabet A. It is easily seen that the relation < is a partial order on ^Y\(A). < is reflexive, it is antisymmetric by the argument above, and transitive since the composition of two homomorphisms onto is again a homomorphism onto. " Definition : An automaton M = (Q, A, f ) defines a congruence m(M) on A* as follows : x rn(M) y ^> f(q,x) = f(q,y) for all q e Q. The finite semigroup A*/m(M) is called the semigroup S(M) of M. -9- It is easily seen that isomorphic automata have the same semigroup, hence we can define an equivalence relation u on "fl^HA) as follows: Definition : M' u M #=> m(M) = m(M'). Several results follow easily from these definitions, "but since they are special cases of later theorems, we will merely mention them in the rest of this section. Let Z(A) be the set of all congruence relations of finite index on A*. By a well-known theorem of Birkhoff [h] , Z(A) is a lattice under the usual partial order of refinement . The function m which associates a congruence on A* with every automaton is a mapping from 7)^ (A) onto Z(A), because for every congruence T of finite index one can find an automaton which has T as its associated congruence (or, equivalently, A*/t as its semigroup). One such automaton is called the automaton of the semigroup k*/i, and is defined ('uniquely up to isomorphism) "by M = (k*/i, A, f), where f(xJT,a) = xaJT is a consistent definition of the transition function by the assumption that T is a congruence. The semigroup of the automaton of a given semigroup A*/ T ls easily seen to be A*/t. An automaton is a semigroup-automaton iff it is the automaton of some finite semigroup A*/r . The equivalence relation m on 711(A) determined by m is order preserving, and jf[(A)/[i is anti-isomorphic to (or isomorphic to the dual of) Z(A), i.e., : -10- 1) yT[(A)/\i is in 1:1 correspondence with Z(A) ("by means of the mapping m) 2) M' |u / MJu •£=> m(M') > m(M) Hence we have the following result: Theorem 1 : In the set ff[(A) of all classes of isomorphic automata on the fixed input alphabet A, partially ordered under homomorphisrns, the equivalence u which identifies automata with the same semi- group is order preserving, and 77\(A)/u is a lattice under the partial order induced hy <, The question arises whether the set 1~f{(A) itself is a lattice under the partial order << The following example shows that this is not the case. The four nonisomorphic automata M , M , M , M, form a cross, i.e., M 3 < M^ M 3 < M 2 , M h o The states of the automata are represented by nodes and the effect of every input symbol a e A by labeled arrows (the labels are omitted in this example since there is only one input symbol). Notice that there can be no automaton M such that M < M , M, < M , ^c S ^-1 > ^c S ^o> since the number of states of any such automaton would have to be > 3 (the number of state of M and of M, ) and < k (the number of states of M and M ). Hence M and M have no greatest lower bound and M and M, have no least upper bound, hence Jf[(A) is not a lattice if A consists of one -12- s ingle element. If A consists of any finite number of elements, consider the four automata in T/1(A) which are just like M^ M^, M , M^ above in the sense that all inputs a e A have the same effect as the single input symbol depicted above. These four automata will still form a cross, hence 7}\(A) is not a lattice for any finite nonempty set A. Notice that S^) = S(M 2 ) = S(M 3 ) e a 2 a e e a 2 a a a 2 a 2 a 2 a 2 a 2 a 2 a while s(u k ) = e a e e a a a a Here the semigroup is described by its multiplication table, and an element of S(M), i.e., a class of input strings, is denoted by any of its representatives. Therefore, in S(M ) above, a denotes the class of all strings of length > 2. We see that modulo the equivalence |i on 7fl(A) the cross is "absorbed" in one class of u, and we have M. I / M, I = M^ I = NL I . Vu l'u 2'u. 3 M- For a deeper study of the equivalence u, particularly of the structure of its equivalence classes, we will consider the lattice of congruences on an automaton M and the lattice of congruences on its semigroup. We will then turn Ion to certain subsets, finite and infinite, of 7V\(A) which are lattices under homomorphism. k CONGRUENCES ON AN AUTOMATON AND ON ITS SEMIGROUP A homomorphism h: M -* M' determines an equivalence relation n on Q as follows : q*q' ^ h(q) = h(q') it is a congruence on M, i.e., q it q' 4 f(q,a) n f (q ! ,a ) for all a e A. For h(q) = h(q«) implies h(f(q,a)) = f(h(q),a) = f'(h(q ! ),a) = h(f(q',a), i.e., f(q,a) rt f(q ! ,a). It is an immediate consequence of the definition of extension of the transition function f that: q it q 1 =^ f(q,x) n f(q',x) for all x € A*. A congruence it on M in turn determines a homomorphic image M/jt = (Q/fl, A,f), where f'(q|n,a) = f(q,a)|it. This definition is consistent, i.e., independent of the representative q chosen, for if q rt q 1 , then f(q,a)|jt = f(q 7 ,a)|:T, "by the assumption that it is a congruence. Let A(m) denote the lattice of congruence relations on M. Although the set of homomorphisms from M (onto some automaton on M ! ) is clearly in 1:1 correspondence with A(M), the set of homomorphic images of M, modulo isomorphism, is not, since it is possible that two distinct congruences it, p, on M give rise to isomorphic quotient automata m/jt = M/p. Definition : Let jt, p be congruences on Mo The equivalence 7 on A(m), called isomorphism, is defined by ■13- -14- jt y p ^ M/rt = M/p <=^ there exists a function h: Q/it -» Q/p , 1:1 onto, such that: h(f(ql«,a)) = f(h(q|it),a) for all q e Q, a e A. Remark : The f on the left-hand side denotes the transition function of m/jt, the one on the right-hand side the transition function of M/p . Since for any congruence n, the transition function f ' of M/it is uniquely determined by the transition function f of M, we will often use the same letter for both, Example M = ({q-L* q 2 ; q^}. (a},f), where f(q,a) = q for all q. M A(M) (q ] _q 2 q 3 ) (cl-j^' q 3 ) (q x ; q 2 a 3 ) (q 2 , q x q 3 ) (q x . q 2 , q 3 ) -15- Here and. in future examples we denote by (q.,q p , qO> e.g., the equivalence relation which has two classes, one containing q and q , the other q . Notice that the three nontrivial congruences above all give rise to the same homomorphic image, namely: M' : By definition of 7, the set jf/l (A) of all homomorphic images of M is in 1:1 correspondence with the set A(M)/7 of equivalence classes of 7. We now want to show that one can define in a natural way a partial order on A(M)/7 (which is induced by the partial order of refinement on A(M)), and that the poset A(.M)/7 is anti-isomorphic to (i.e., isomorphic to the dual of) 7H m (a). Theorem 2 : The equivalence 7 on A(M) is order preserving (see Appendix, p.^5) c Proof: First, notice that it < p e A(M) implies: #(n) = (number of classes in n) > (number of classes in p) = #(p) with equality holding only if jt = p. Next, consider a closed stair (Appendix, p. ^5) rt n p., . . • Jt , p with respect to 7 in A(M), i.e., jt. , < p., p. 7 Jt. and p 7 jt„. This implies • ' l-l — K i' K i ' 1 "n • #(* ) >#(p ) =#(n x ) > ... #( Vl ) >#(p n ) = #(jt Q ), -16- hence that all congruences in the closed stair have the same number of blocks. Hence every inequality rt. , < p. must be an equality re . _ = p., hence the whole stair lies entirely within one class of 7. By Theorem 1 of the Appendix, the stair relation / on A(M)/V induced by < on A(M) is a partial order. Theorem 3 • The equivalence 7 on A(M) is directed upward (DU, see Appendix P- 53). Proof : Let jt, jt 1 , p' e A(M) be such that it' < jt and it' 7 p', i.e., there exists an isomorphism h, m/jt' -*M/p', h(f (q | n 1 ,a) ) = f (h(q |n ! ) ,a) . Since n' is a refinement of it, h maps n into some equivalence p, which is clearly > p' First, p is a congruence,, because q p q' entails the following series of equalities : q|p h _1 (q|p ! ) f(h _1 (q|p'),a) h" 1 (f(q|p',a)) f(q|p',a) f(q,a)|p« f(q,a) p/p jr/jt it/it n/it p/p p/p P q'lp' h" X (q'|p') f(h" 1 (q'|p , ),a) h" 1 (f(q'|p ! ,a)) f(q' |p',a) f(q',a)|p' f(q',a) (h " is an isomorphism) ( Tt/rc ! is a congruence on M/n') (h is an isomorphism) (h is an isomorphism) (p/p 1 is a congruence) Second, n 7 p, since the isomorphism h: M/V -> M/p ' obviously remains an somorphism if considered as a mapping m/tt -» M/p. Hence the congruence p exhibited here has the properties required in the definition of DU. -17- This yields another proof of Theorem 2 by invoking Theorem 9 of the Appendix (DU =3> OP). Next we want to show that if M" = M A " < M' = M/it ' e 7fl M (A), then it ! \y f it " \y . For M" < M' implies that there is a congruence p on M' such that M" = M'/p = (M/it')/p. Now (M/it ' )/p determines a congruence p' on M and p 1 > it ' . And moreover, M" = M/it " = M/p ' =£ it" 7 ft' > hence it ' < p ' 7 « " is a stair from it ' | 7 to it " \y . Second, if it 1 1 7 f it" 1 7 then M" < M' e TTl (A) . For 7 is directed upward by Theorem 3 and this implies (see Appendix, p. 53) that there exist congruences p', p" on M such that jt 1 7 p 1 < p" 7 it'. Then M" = M/p" = (M/p ! )/(p"/p ! ) = M'/(p'7p') ? i-e., M" is a homomorphic image of M* . Thus we have proved the following Theorem k : The set 7' I (A) of homomorphic images of a given automaton M, under the partial order < defined by homomorphism, is isomorphic to the dual of the poset A(M)/y, where A(M) is the lattice of congruences on M and 7 the equivalence which identifies isomorphic congruences . This reduces the study of the structure of Jf\ (A) to the study of the equivalence 7 on A(M) . Let Z(M) denote the lattice of congruence relations on the semigroup S(M) = A*/m(M) of M. -IB- Definition: The associated congruence s(n) on S(M) of a congruence it on M = (Q, A, f ) is defined by: x s(it) y iff for all q e Q, [f(q,x) it f(q ? y)]« Here x and y denote elements of S(M), i.e., classes of input strings, rather than single input strings. f(q,x), of course, is defined by taking any representative string from its class . It is easily seen that the following statements are equivalent to this definition: 1) x s(it) y O ( V q,q' e Q) [q n q' =^ f(q,x) « f(q',y)] 2) x s(jt) y <^> ( \/ (Vq) [h(f(q|n,x))= h(f(q|jT,y))] 4=4> ( Vq) [f(h(q|4x) = f(h(q|n),y)] and since h is 1:1 onto, <^> ( Vq') [fOi'lp,*) = f(q'|p,y)] ^=* x s(p) y Theorem 6 : If -a < p e A(M), then s(jt) < s(p), i.e., the mapping s: A(M) -» E(M) is monotonic . Proof: x s(jt) y <£^> ( Vq) [f(q,x) jt f(q,y)] <£=^ ( Vq.) [f(q,x) p f(q,y)] O x s (p) F- This result cannot be strengthened to n < p e A(M) =^ s(it) < s(p), as will be seen. Given a congruence T on S(M), there may or may not exist a congruence ir on M such that s(jt) = T. An example where there exists no such jt is the following: M -20- S(M) e b a ab e e b a ab b b e a ab a a ab a ab ab ab a a ab A(M) consists only of the two trivial congruences 1_ = (q, q ? ) and n = (o.-i ) 1 ) ) whose associated congruences on S(m) are obviously 1 = (e a b ab) and = (e, a, b, ab). However, there are two more congruences on S(M), namely s T = (e, b, a ab) and T» = (e b, a ab), and these are not associated with any congruence on M. There are congruences T on S(M) such that S(M)/t is the semigroup of no homomorphic image of M; of course S(m)/t is the semigroup of at least one automaton, namely the automaton of S(m)/t, but this need not be a homomorphic image of M. However, for every congruence T on S(M) we can define a certain congruence s(t) on M which is of particular importance. Definition: 1) q s(T) q' if (3 q Q )( 3 x)(3 x')[q = f(q Q , x), q' = f(q Q , x') and x t x ' ] 2) "s(t) is the transitive closure of the relation that holds between pairs (q,q') by l). -21- The verification that s(t) is indeed a congruence is straightforward Notice that in the last example, s(t ) = s(t ) = 1 . J. d. m Theorem 7 : If T 1 < T on S(M), then s(t') < s(t). Proof: Let l) (t) denote the relation on Q defined in l) in the above definition. In order to prove the theorem it is clearly sufficient to show that if l) (t 1 ) holds between q and q', then l) (t) also holds. The same will then be true for s(t') and s(t), which are the transitive closures of 1)(t') and 1)(t) respectively. Now q 1)(t') q 1 <£-> ( 3 q Q )( 3 x)( 3 x')[q = f(q Q ,x), q' = f (q Q ,x' ) and x t" x'] =* ( 3 1 )( 3 x)( 3 x')[q = f(q Q ,x), q' = f(q Q , x'), and x T x'] 4=p q l)(i ) q' . Theorem 8 : s(s(jt)) < n for all congruences it on M, Proof : Again, it is sufficient to show that l)(s(it)) < irj then the transitive closures of these two relations which are s(s(jt)) and it respectively, satisfy the same inequality. Now ( 3 q Q )( 3 x)( 3 x')[q = f(q Q , x), q' = f(q Q , X 1 ) and x s(it) x'], where x s(*) x* 4=$ ( V p)[f(p,x) « f(p,x')], implies q = f(q Q , x) rt f(q Q , x') = q', i.e., q it q' . -22- Theorem 9 : s(s(t)) > t for all congruences T on S(M), and equality holds iff T = s(n) for some congruence n on M. Proof: The first part is a mere restatement of definitions: x T x' =*► ( V q)( 3 q Q )( 3 z)( 3 z')[f(q,x) = f(q Q ,z), f(q,x' ) = f(q Q ,z' ), and z T z ' ] £^ (V q)[f(q,x) J(t) f(q,x')] * s(s(t)) x' . Second, if T = s(jt) for some congruence jt on M, then s(n) < n by Theorem 8, hence s(s(t)) < s(it) = T by Theorem 5- But we have just proved that s(s(t)) > 1, hence s(s(t)) = T. Third, if s(s(t)) = T, then T is trivially the associated congruence of s(f). An immediate consequence of the last two theorems is Theorem 10 : If i = s(jt) for some congruence it on M, then among all the congruences n' which have T = s(jt') as their associate, there is a least one, namely s(t). In terms of automata this means that for a given quotient semigroup S(M)/t, if there is any automaton M' < M such that S(M' ) = S(m)/t, then there is a greatest one (in 7T\ (A)). The following example shows that in general there is no greatest mgruence among all the congruences on M which have a given congruence T on M) associated with them. -23- M: S(M) e a e e a a a a A(M) £(M) 1 Q = (q x qg qJ = (q 1 , qg-, q ) (q x , q 2 qJ (q 2 > q x qJ (q 3 > q x qg) l s = (e a) O s = (e, a) s(l ) = 1 , while the other four congruences in A(M) have as their associate. Notice that the three nontrivial congruences in A(M) are isomorphic, and correspond to the homomorphic image M ? : Consider now the set ^ n (M) of those congruences on S(M) which are associated with some congruence n on M. -2k- Theorem 11 : s(« • n') = s(jt) • s(jt'), and (hence) if t, T 1 e Z (M), then T . T' 6 Z Q (M). Proof : x b(jc) . s(*») y ^=* ( Vq)[f(q,x) it f(q,y) and ( V q)[f(q,x) *' f(q,y)] *=> ( V q)[f(q,x) « • *' f(q,y)] <#=^ x s(jt • «' ) y. Theorem 12 : s(jt + Jt') > s(n) + s(rt'), and equality does not hold in general, Proof : The first part of the theorem is an immediate consequence of the previous result n < p =^ s(rt) < s(p). The more interesting second part is proved by the following example : -25- Consider the two congruences n = (l 3; ^ 5> 2) and p = (l 5, 2 3, k) , and their l.u.b. it + p = l Q = (1 2 3 ^ 5) Denoting elements of S(m), i.e., classes of input strings, by representatives we find S(M) = {e, a, b, c, c , ab, ac, be}. Now s(jt) is the congruence on S(m) generated by e = a, and s(p) the one generated by e = b. Thus s(jt) + s(p) is generated by e = a = b, and this is f the congruence 1 which identifies all elements of S(m), since c and e, e.g., still are in different classes of s(n) + s(p). On the other hand s(:r + p) = s(l) = 1, hence s(n) + s(p) < s(jt + p) for this example. Theorem 13 : Z (M), the set of congruences on S(M) which are associated with some congruence on M, is a lattice, but in general not a sub- lattice of Z(M). Proof : If t 1 , T 11 e Z (M), then there exist a g.l.b. T 1 • t" e Z (M), according to the previous theorem. Consider the set of all congruences T e Z (M) which are > both T 1 and t". This set is not empty since it contains 1 = s(l ), hence by Theorem 11 it has a g.l.b. which is again in Z (M), and S loi u this g.l.b. is easily seen to be the l.u.b. of t' and t" in Z (M) . That Z (M) is in general not a sublattice of Z(M) is seen by the last example, where the l.u.b (as operation in Z(M)!) of two congruences in Z (M) is not in Z (M) . -26- If we define an equivalence a on A(M) by fl op <-=,> s(it) = s(p) then we know by Theorem 5 that 7 < a, i.e., the equivalence isomorphism is a refinement of a, and hence by (A(M)/V) = Til (A) (Theorem h) , that (A(M)/a) 7'L (A)/u, where u is the equivalence on W» (A) which identifies automata with the same semigroup. On the other hand, we have just seen that the function s : A(M)/a -*■ ^ n (M) is an isomorphism, hence we obtain Theorem 13 : In the set TH (A) of all homomorphic images of M, modulo isomorphism, the equivalence u which identifies automata with the same semigroup is order preserving, and \i\ (A)/u is a lattice under the partial order / induced by <• In conjunction with the easily proven lemma: if M' < M, then 77\ ,(A)/u- is a sublattice of 7V\ m (A)/u, one obtains Theorem 1 which was anticipated in Sec. 3« Our main interest, however, is to find conditions on M which imply that m (A) itself is a lattice. The next two sections are devoted to this . Theorem 13 can be interpreted in a different way. By Theorem 10, we can associate with every u-class on ''V (A) a canonical automaton, namely the greatest in its class (corresponding to the least congruence in a a-class on A(m)). Two such canonical automata are related by < iff their respective classes are related by /. Hence we find: -27- Theorem: For any automaton M, the subset of jTIivXa) consisting of all automata which are the greatest in their u-class is a lattice under homomorphism. 5- PROPERTIES PRESERVED UNDER HOMOMORPHISM Let P be any property of automata which is preserved under homomorphism; i.e., if M satisfies P and M' < M, then M' also satisfies P. Everything we have proved so far holds if we replace T/((A) by Til (A), the set of all automata with property P, with the obvious changes of interpretation. Notice in particular that for any M satisfying P, i)n M (A)=Tfn*(A). We will later make use of a theorem of Lyndon [5]: A property is preserved under homomorphism iff it can be expressed by a positive sentence, i.e., a sentence of the lower predicate calculus, which contains only AND and OR as connectives. As an example, strong connectedness is preserved under homomorphism, since it can be defined thus : ( V q) (V q»)( 3 x)[f(q,x) = q'] Definition: An automaton M satisfies property T iff for every congruence it on M, S(M/jt) = S(M) =#■ it = (i.e., every quotient automaton m/jt, except the one defined by the zero congruence, has a strictly smaller semigroup than M has . -28- -29- Property T is not preserved under homomorphism. Since it is in general not so easy to prove that a certain statement cannot be expressed "by a positive sentence as it is to exhibit such a sentence if it exists, we will not make use of Lyndon's theorem here but merely give a counter- example . Consider: M: whose semigroup S(m) = Z^, the cyclic group of order 6. All the nontrivial congruences (l, 2, 3> ^ 5)> (l 2 3j **-, 5), (l 2 3, ^5) give rise to smaller semigroups, namely Z , Z , Z respectively, hence M satisfies property T. However, M/(l 2 3> 5 5) is the automaton; which does not have property T, since it and its homomorphic image both have Z as their semigroup , -30- Theorem lk: Let the automaton M satisfy a property P which is preserved under homomorphism and which implies T. Then, on the lattice A(M) of congruence relations on M, a - j - 0, i.e., the equivalence defined by having the same associated congruence and the equivalence isomorphism both degenerate to the zero equivalence. Proof : Let jr', rt" e A(M) and n' a it". Then by Theorem 9 > there exists jt e A(M) such that jt < jr' • jt" and n a jr 1 (e.g., s(s(jt')) is such a congruence jt). Consider m/jt and its homomorphic image m/jt' = (m/jt)/(jt '/ Jt ) . Since P is preserved under homomorphism, m/jt satisfies P and hence T. On the other hand, since s(jt) = s(tt'), S(m/jt) = S(m/jt'), from which it follows that jt 1 /jt = 0, or jt = jt' . Similarly, jt = jt", hence jt' o jt" implies jt' = jt", or 0=0. And since a > J by Theorem 5 > we also have 7=0. An immediate consequence of this and Theorem k is : Theorem 15 : Let M satisfy a property P which is preserved under homomorphism and which implies T. Then 7/1 (A) is a lattice. p In order to extend this result to the set fjf\ (A) of all automata satisfying P, we have to make an additional assumption. Definition : P is an upper bound property iff for all M', M" satisfying P, there exists an M satisfying P such that M' < M, M" < M. -31- Theorem l6 : If P is 1) preserved under homomorphism 2) an upper bound property, and 3) P implies T, p then the poset Tfl(\ (A) of all finite automata on A satisfying P is a lattice. Proof: Let M , IVL be two automata in "f(\ (A) (exclude the trivial case where no automaton satisfies P). There exists at least one upper bound in p yn (A) by assumption, and a lower bound, namely the one-state automaton which is a homomorphic image of M and M and hence also satisfies P. Assume there p were two upper bounds M , M, in Yf\ (A) which were minimal, and choose an upper bound M for M and M, which also satisfies P. By Theorem 15, Iff] (A) = 7]f[ (A) is a lattice, hence there has to be an automaton IVL, satisfying P, such that M -i S M c < M o and M P < M s S M k^ contradicting the assumption that M , M, were minimal upper bounds. Hence there exists a l.u.b. of M, and M in Jl\ (A). A similar argument holds for the g.l.b. That P must be an upper bound property for this theorem to hold will be seen in the next section, where we give an example of a property P such that for all M satisfying P, Tf[ (A) is a lattice, yet 7n ( a ) is not * The next result is of interest in connection with Krohn and Rhodes' theory [3] of series -parallel realization of automata. The building blocks of such a realization are automata with a given semigroup. If we restrict ourselves to realizations out of automata satisfying a property P as above, then these building blocks become essentially unique. -32- Theorem 17 : Let P be any upper bound property which is preserved under homomorphism and implies T. Let T be any congruence of finite index on A*. If there exists any automaton M satisfying P such that S(M) = k*/i , then it is unique up to isomorphism. Proof : Let M 1 , M" be automata satisfying P such that S(M' ) = S(M") = k*/i, and choose an upper bound M of M' and M' which also satisfies P. If n' , si" are congruences on M such that M' = M/V , M" = M/V', then S(M' ) = S(M)/s(it' ), S(M") = S(M)/s(V')and hence s(it') = s(jt")j or it" a it". By Theorem lk, a = on A(M), hence n' = it", hence M' = M" . 6. CERTAIN LATTICES OF AUTOMATA In this section we investigate certain specific classes of automata which satisfy a property P of the kind considered in Sec. 5« Definition : M is a single component automaton iff there exists a state q e Q such that, for all q e Q, there exists an input string x € A* with f (q ,x) = q. Definition : M is strongly connected iff for all q,q'e Q, there exists an x € A* such that f(q,x) = q 1 . Definition : M is abelian [6] iff for all q e Q, and for all a, t> e A, f(q, ab) = f(q, ba). Definition : M is a reset automaton iff there exists q € Q and x e A* such that for all q e Q, f(q ; x) = q • Definition: M is circular iff there exists x e A* such that for all q,q' e Q there exists an integer k > with the property f(q,x k ) - q'. It is easy to verify that an automaton is circular iff there exists an input string whose picture in the state graph is a closed loop which includes all states. The reason for formulating this definition as it stands is that in this form it is immediately apparent that it will result in a positive ■33- -3*- sentence when formalized. The same is true of the other four definitions , hence it follows immediately from Lyndon's Theorem on page 28 (and from the fact that the conjunction of two properties preserved under homomorphism is again preserved under homomorphism) that: Theorem 18 : Each of the properties: 1) single component abelian 2) strongly connected reset automaton 3) circular is preserved under homomorphism. Theorem 19: Each of the properties : 1) single component abelian 2) strongly connected reset automaton 3) circular implies property T. Proof : Let tt be any congruence on the single component abelian automaton M. Since S(m/jt) = S(M)/s(jt), we have to show that s(jt) = implies it = 0. Let q jt q', q = f(q Q , x) and q' = f(q Q , x'). Let p be any state in Q, and let P = f(q Q ,y). Then -35- f(p,x) = f(q Q , y x) = f(q Q , x y) = f(q,y) ix f(q',y) = f(q Q , x'y) = f(q Q , y x 1 ) = f(p,x ! ), therefore, f(p,x) it f(p,x') for all p 6 Q, or x s(it) x'. If a (it) = 0, then X = x 1 , hence q = q' , hence it = 0. 2) If M is a strongly connected reset automaton, then clearly for any state q there exists a reset input string x (i.e., an x e A* such that for all q' e Q, f(cL», x) = q). Let q it q 1 , f(q,z ! ) = q ? , and let f(p,z) = q for all p e Q. Then we have f(p,zz') = q' for all p e Q, hence for all p e Q, f(p,z) it f(p, zz'), or z s(it) zz ! . If s(ft) = 0, then z = zz', hence q = q : , hence it = 0. 3) Let M be circular, and x € A* the input string whose existence is k required "by the definition- Assume q it q s , and let f(q,x ) = q' . Let p "be any state, and let f(q,x ) = p. Then q it q ! implies f (q,x ) it f(q 8 ,x ) = f(q, x ) = f(p,x ), or p it f(p,x ). Hence q it f(q,x ) implies p it f(p,x ) for all p e Q,, hence x s(it) e. If s(it) = 0, then x = e ; hence q = q", hence it - 0. From Theorems 18, 19 and 15 it follows immediately Theorem 2.0 : If M satisfies any one of the following properties: 1) single component abelian 2) strongly connected reset automaton 3) circular then the set 7YI (A) of homomorphic images of M is a lattice. -36- " | /I (A) is a lattice for every M satisfying a property P" does p not imply that YY[ (A), the set of all automata with property P, is a lattice. As an example, consider the property "M is circular": Let *'i 1^ I ) \ iJl M": We want to show that there is no upper bound for M' and M" in the set of all circular automata. Assume M is a circular automaton, it' , it" two congruences on M such that M/V = M ! and M/it" = M" . Let it = it' • it". Then m/jt is also an upper bound for M' and M", and since circularity is preserved under homomorphism, M/jt is circular. Moreover, since M' and M" both have two states, it' and it" have two equivalence classes each, hence it = it ! 'it" has either three or four equivalence classes (not two because this would imply it' = it", hence M' = M"). Hence, if there is a circular automaton which is an upper bound for M' and M", then there is one with either three or four states . Now it is easily seen that the number of states of any homomorphic image of a circular automaton divides the number of states of that automaton (which rules out a three-state automaton), and that a four-state circular automaton has only one two-state homomorphic image (only one congruence relation with two classes). This rules out the possibility that a four-state circular automaton is an upper bound for both M' and M" . Hence the set of all ■j.ta is not a lattice. -37- In contrast to this, we have the following result: Theorem 21 : "Single component and abelian" is an upper bound property, hence ("by Theorems 18, 19, 16) the set of all single component abelian automata is a lattice. Proof: Let M', M n be two single component abelian automata, q 1 , q" two states in M' , M" respectively from which every other state in M f , M" respectively is accessible. (M',q ( ) determines a right invariant equivalence relation [7] 0' on A* by the definition x f y iff f'(q',x) = f'(q',y)o Tnen x 0' y implies xz 0' yz for all z e A*. Now M' = (Q', A, f 1 ) is isomorphic to (A*/0* , A, cp'), where the transition function cp' is defined by cp'(xJ0',a) = xa|0'. If 0" is the right invariant equivalence associated with (M",q"), then = f . 0" is a right invariant equivalence and (A*/0, A, cp) is a single component automaton which is an upper bound for M ! and M", It is also abelian, because cp(u|0, xy) = uxy|0 = uxy|0' n uxyj©" = cp ! (u|e f , xy) fl cp"(u|e", xy) = X 9'(u|0 ! , yx) D cp" (u|0", yx) = uyx|0' fl uyx|0" = uyx|e = cp(u|e, yx). The two automata M e , M" on page 36 have no upper bound in the set of circular automata, but they do have one in the set of strongly connected automata, e.g., their parallel connection M° X M". This raises the question whether or not the set of all strongly connected automata is a lattice or not. A counter- example found by D. E. Muller shows that this is not the case. 1 Because (k*/e* , A, y ! ) = M 1 and M ! is abelian, 7. THE PARTIAL ORDER DEFINED BY SUBAUTOMATA Definition : A closed subset Q' of an automaton M = (Q, A, f ) is a set Q' c Q such that for all q e Q' and for all a e A, f (q,a) e Q 1 . The empty set is considered to be a closed subset. A closed subset Q 1 determines a subautomaton M' = (Q 1 , A, f), where f ' is the restriction of f to Q' x A. Definition : M' c= M, or M' is a sub image of M, iff M' is isomorphic to a subautomaton of M. Sometimes a different concept of subautomaton is used which requires Q' to be closed only under a subset A' of the set A of inputs. Since we are dealing with the set of automata on a fixed input alphabet, such a (generalized) subautomaton of an automaton in Ti\(a) would not be in Kjj(A) and hence without interest for our purpose. The reason for distinguishing subautomata (or closed subsets) and subimages is the same as for distinguishing congruences on and homomorphic images of an automaton: with our convention of identifying isomorphic automata, distinct closed subsets may give rise to isomorphic subautomata and hence to the same subimage. For the present purpose it is convenient to introduce an empty automaton (whose set of states is empty). Hence in this section |f[(A) will denote the set of all classes of isomorphic finite automata including the class of the empty automaton. -38- -39- Tt is easily seen that jl^ ( A ) is partially ordered by c. We will "briefly sketch how results about this partial order can be obtained which are completely analogous to those of the previous section for <• If Q', Q" are closed subsets of an automaton M, then Q' H Q" and Q' U Q" are also, hence the set l(M) of all closed subsets of an automaton M is a lattice under set inclusion. Again we can define the equivalence 7 of isomorphism on L(M) by Q' 7 Q !I ^ (Q«, A, f) = (Q", A,f") where f 1 , f" are the restrictions of f to Q' x A and Q" x A respectively. One can show from these definitions that > is order preserving and directed downward and that m M (A) = L(M)/ 7 where the partial order on the left is that of subimage, and the one on the right the stair relation induced by set inclusion, 0?l (A) now denotes the set of all sublmages (notice: not subautomata! ) of M. With every closed subset Q' of M we can associate a congruence s(Q') on the semigroup S(M) as follows: x s(Q ! ) y 4£$ f(q,x) = f(q,y) for all q e Q 1 . In particular we associate the congruence which identifies all elements of S(M) to the empty closed subset. Conversely, for every congruence T on S(M) we can define a closed subset s"(t) on M by: q e s"(t) iff for all x,y: x t y =p f(q,x) = f(q,y) 40- It is now a matter of verification that statements which are in complete analogy to Theorems 5 through 13 of Sec. k hold. These are obtained from the Theorems of Sec. h by replacing every occurrence of "congruence on M" by "closed subset of M", every occurrence of "<" between congruences on M (not between congruences on S(M)!) by "3" (notice that the direction is reversed.'), every occurrence of "." and "+" between congruences on M by "U" and "fl" respectively, and every occurrence of "homomorphic image" by "subimage" For example, Theorem 5: jt 7 p =$> s(]t) = s(p) and Theorem 6: n < p =p s(jt) < s(p) become while 5«): Q- 7 Q" => s(Q') = s(Q") and 6 ! ): Q ! 3 Q» =>> S (Q') < S (Q»), Theorem 11: s(n • « ' ) = s («) • s U ' ) Theorem 12: s(n + rt « ) > s ( it) + s(it ' ) and become 11'): s(Q' U Q") = s(Q') ■ s(Q") 12'): s(Q« fl Q") > s(') + s(Q") and -in- Furthermore, since the proof of the theorems in See. 5 depended merely on the results of Sec. ^-, analogous statements to the whole of Sec. 5 will hold if we redefine T thus 1 An automaton M = (Q, A, f) satisfies property T iff for all closed subsets Q' of M, s(Q') = s(Q.) =^> Q," = Q and replace "P is a property preserved under homomorphism" everywhere by "p is a property preserved under passage from automaton to subautomaton". Unfortunately, finding such properties P which are interesting and intuitively meaningful seems to be more difficult in the case of subautomata Clearly anything involving strong connectedness is uninteresting, since a strongly connected automaton has only trivial subautomata--itself and the empty automaton. However, it is interesting that one can establish such a close analogy between two so seemingly unrelated concepts as homomorphic image and subautomaton . 8. CONCLUDING REMARKS We have studied the structure of certain sets of finite automata under two partial orders, defined by homomorphisms and subautomata respectively. The motivation for this work, namely, as a step towards imposing more general algebraic structures on sets of finite automata, was explained in the introduction. We would like to conclude by mentioning some applications of and problems related to the present investigation. There is a close relation between the partial order "homomorphic image" and loop-free decomposition of automata: The front component of a series decomposition and any component of a parallel decomposition is a homomorphic image of the automaton to be realized, if one restricts oneself to decompositions determined by partitions with the substitution property [1,2], or in any case a homomorphic image of the automaton defined by the loop-free structure. Assume we are interested in decompositions in which the total number of component automata is not as important as the number of different types of components (i.e., two component automata which are isomorphic are counted as one. In a technology such as integrated circuits, where additional copies of a given circuit have a low cost once the first copy has been obtained, such an assumption is realistic). It would seem that our formalism and results can be applied to several questions in this area. For example, if there exist two congruences n, n' on an automaton M such that n • n' =0 and it y jt" , then M can be decomposed into a parallel connection of two copies of the automaton m/tc. For certain classes of automata such as those discussed in Sec. 6 which have / 0, such a decomposition cannot exist. One would then have to take recourse ■1+2- -k3- to state splitting (or equivalently, to decompositions using set systems [8]),, in such a way as to get a larger automaton with y ^ 0„ Loop-free decomposition with a minimum of different types of component automata poses many problems which have never received attention „ Let us mention a possible generalization of the present investigation which we have actively but unsuccessfully pursured. All the results on the structure of Y/l (A) were derived by setting up a correspondence between f}/[ (A) (or some subset of it) and the lattice of all finite semigroups of the form A*/t (or some subset of it which is a lattice),, The question is whether the lattice of semigroups can profitably be replaced by some other algebraic system--where "profitably" means that it imparts a rich structure to jni.A), The answer to this question is necessarily subjective,, All we can say is that several attempts in this direction failed to yield any results 9- BIBLIOGRAPHY [1] J. Hartmanis, "Symbolic Analysis of a Decomposition of Information Processing Machines," Information and Control, Vol. 3; No. 2, pp. 15U-178; June, i960. [2] J. Hartmanis, "Loop-Free Structure of Sequential Machines," Information and Control, Vol. 5, No. 1, pp. 25-^3; March, 1962. [3] K. B. Krohn and J. L. Rhodes, "Algebraic Theory of Machines," Proceedings of the Symposium on Mathematical Theory of Automata, Polytechnic Institute of Brooklyn, N. Y., pp. 3^1-384; 1962. [k] G. Birkhoff, "Lattice Theory," AMS Publication, 19^8. [5] R. C. Lyndon, "Properties Preserved Under Homomorphism, " Pacific J. of Math., Vol. 9, No. 1, pp. 143-154; 1959- [6] A. C. Fleck, "isomorphism Groups of Automata," Journal ACM, Vol. 9; No. k, pp. 469-U76; October, 1962 . [7] M. 0. Rabin and D. Scott, "Finite Automata and Their Decision Problems," IBM J. of Research and Development, Vol. 3, No. 2, pp. 1Xk-12k; April, 1959. J. Hartmanis and R. E. Stearns, "Pair Algebra and its Application to Automata Theory," Information and Control, Vol. 7; No. k, pp. 485-507; December, 196^. ■hk- APPENDIX ORDER PRESERVING EQUIVALENCE RELATIONS IN PARTIALLY ORDERED SETS Let P be a finite set, partially ordered by the relation <, and let 7 be an equivalence relation on P. < determines a relation f' on the set P/7 of equivalence classes of 7 as follows: Definition: A sequence p~q, p,q P~ • . • P , Q_ , P. » 1. e P> is called an ■ Oil 2 2 n-1 tt 1' 1 ' ascending stair (with respect to 7) from the equivalence class p J 7 of p to the class ql 7 of q iff p. < q. , i = 1, „.. n and q. 7 p., i = 1, ..., n-1. The existence of an ascending stair from p | 7 to q_ j 7 is denoted by p | 7 f q| 7 . f is called the stair relation on P/7 induced by the partial order < on P. Definition : A stair p q p . .. p q is closed iff p 7a . The sequence pp is a closed stair with respect to any equivalence relation. Definition ; An equivalence 7 in a poset P is order preserving, or OP, iff every closed stair (with respect to 7) lies entirely within one class of 7. Theorem 1 : 7 is order preserving in P iff the stair relation f on P/7 induced by < is a partial order. -U 5 - -U6- Proof => : 1) f is reflexive, because for all p e P, pp is a stair from pjy to pJ7, hence p | 7 f p \y . 2) f is transitive: if P q.-,P-, • • • ^ and Pa ^ Pi • • • P' from a poset into another is monotonic (or an order homomorphism) iff p < q =$, f(p) < f(q). Theorem 2 : If 7 is an OP equivalence on P, then the natural mapping f : P — » P/7 which maps every element into its equivalence class is monotonic . Proof: If p < q then pq is a stair from the class p | 7 to the class q|7, hence pi 7 f q 1 7. -vjr- Theorem 3 : The natural equivalence 7 on a poset P, defined by a monotonic function f: P -+ P 1 as follows: p 7 q iff f(p) = f(q), is OP. Proof : Let p o ... q be a closed stair with respect to 7. Since f is monotonic, we have f(p ) hence the stair lies within one block of 7. We see that OP equivalences on a poset play the same role as congruence relations on algebras. This analogy is strengthened by the following results on the set of all OP-equivalences on a poset. Theorem k : If 7,0 are OP equivalences on a poset P, then their g.l.b. 7*8 is also OP. Proof : Let p q p ... q be a closed stair with respect to 7-8. Then this sequence is a stair both with respect to 7 and 5, which are OP equivalences, hence the sequence lies in one class of both 7 and 5, hence in one class of 7 •& . Notice that the l.u.b. 7 + 8 of two OP-equivalences 7 and 8 is in general not OP. An example is: 1 7 = (p s, q, r, 0, l) is OP 8 = (p, s, q~7, 0, l) is OP I 7 + 8 = (p s, q r, 0, l) is not OP -48- Definition ; If 7 is any equivalence on a poset P, then 7, the OP equivalence generated by 7, is defined as follows: p ~y q iff p I 7 f q_\y and q. | 7 / p | 7 7 is indeed OP. If Pp. 1 !-.?-! ... P , Q is a closed stair with respect to y, then for any p. in this sequence, p | 7 f p. | 7 and p. | 7 f p | 7, i.e., p^ ~y p. for all p., hence all p. and all q. lie in the same block of 7. ' ' 1 1 1 1 Clearly 7 > 7. Theorem 5 : If t] is an OP equivalence, 7 any equivalence, and r\ > y, then t) >y. Proof : Let p 7 q, i.e., by definition, p| 7 f q| 7 and q_ 1 7 / p 1 7 . Then t] > 7 implies p | rj f q. | rj and q. | rj f p|r|; the concatenation of the two stairs whose existence is asserted by those statements is a closed stair with respect to T) which contains, among others, the two elements p and q. Since r\ is OP, this stair lies entirely within one class of r\, and in particular p T] q. Hence r\ > 7. Theorem 6 : The set of all OP equivalences on a poset P is a lattice under the usual partial order of refinement for equivalence relations If 7,6 are OP equivalences, their g.l.b. in this lattice is 7«5, and their l.u.b. is 7 + & (where • and + denote the g.l.b. and l.u.b. in the lattice of all equivalence relations on P). The proof is an easy consequence of Theorems h and 5. Notice that of all OP equivalences on P is a subset of the lattice of all equivalences on P which is a lattice under the same partial order, but is not ^lattice since the l.u.b. is defined differently. -1*9- Definition : An equivalence 7 on a poset P is cross-free, or CF, iff for any classes A, B, C, D of 7 such that A f C, A f L, B f C, B f D there exists a class E of 7 such that A f E, B f E, E f C, E f D. OP and CF are independent properties. The following two examples show equivalence relations on lattices which satisfy one but not the other: OP but not CF CF but not OP The equivalence is indicated by encircling all elements belonging to the same class. On a poset L which is a lattice, an OP equivalence 7 defines a poset L/7 which may or may not be a lattice under f. The example on the left above (OP but not CF) is a case where the quotient L/7 is not a lattice -50- If 7 is the equivalence which identifies all elements of L, then L/7 is trivially a lattice. Theorem 7 : If "the equivalence 7 in a lattice L is OP, then: L/7 is a lattice ^-fr 7 is CF. Proof =» : Consider four classes A, B, C, D of 7 such that A f C, B f C, A f D, B /* D. Since L/7 is a lattice, either the g.l.b. of C and D or the l.u.b. of A and B is an intermediate class E as required by the definition of CF. Proof 4= : If A and B are classes of y, consider the set S of all classes F such that A f F, B f F. From the assumption that L is a lattice it follows immediately that S is not empty. Moreover, there must be a least (unique minimal) element of S. For assume C, D were two minimal classes. Then we would have A f C, A f D, B f C, B f D. Since 7 is CF, there would have to be a class E such that A f E, B f E, E f C, E f D, contradicting the assumption that C and D were minimal in S. The dual argument shows that there is a greatest class in the set of all classes which are f A and / B. Hence L/7 is a lattice. One of our main purposes is a study of lattices from a point of view which stresses the order properties rather than the algebraic ones based on the two binary operations g.l.b. and l.u.b. As a consequence, in place of the important concepts of (operation-) homomorphism, congruence relation and sublattice which arise naturally when lattices are considered as algebras with two binary operations, we have various concepts of monotonic ■ f ,ion (or order homomorphism), OP and CF equivalence relation, and "subset is a lattice under the same partial order", respectively. -51- In order to get a complete analogy, however, the concept of monotonic function is too weak. For although we can prove: If 7 is an OP and CF equivalence on a lattice L, then L/y is a lattice under the stair relation and the natural mapping L — > L/7 is monotonic, the companion statement to this theorem is not correct. The natural equivalence 7 defined by a monotonic function f from a lattice L onto a lattice L' by: p 7 q iff f(p) = f(q), is OP (by Theorem 3), but in general not CF. Example e' = f ■ -> L' -52- The difficulty lies in the fact that monotonicity alone imposes no restriction on the images of two incomparable elements. Definition : A function f : P -» P' from a poset into another is strongly monotonic iff 1) p < q => f(p) < f(q), and 2) f(p) < f(q) =^ p j 7 f q| 7 , where 7 is the natural equivalence defined by f . Theorem 8 : l) If 7 is an OP and CF equivalence on a lattice L, then L/7 is a lattice under the stair relation and the natural mapping f :L -» L/7 is strongly monotonic. 2) The natural equivalence 7 defined by a strongly monotonic function from a lattice L onto a lattice L' is OP and CF, and L/ 7 * L- . Proof : 1) is an immediate consequence of Theorem 7 , Theorem 2 and the definition of strong monotonicity. 2) by Theorem 3, 7 is OP. Let p | 7 f r\y , v\y f s \y, l| 7 f r \j> <1 1 7 / s | 7 in L/7, and let p', q' , r ! , s 1 be the images of p, q, r, s respectively under f. Since f is monotonic, p' C r' , p' f(p) < f(q), hence L/7 3 L' . -53- Notice that the theorem is not true in general if f is a strongly monotonic function into a lattice L' . The order concepts introduced above are generalizations of the corresponding algebraic concepts in the case of lattices: a (lattice-) horaomorphism is strongly monotonic and a congruence relation on a lattice is OP and CF, while the converse is not true. The following equivalence 7 on the lattice L is OP and CF but is not a congruence: L/ 7 : The natural mapping L -» h/y is strongly monotonic but is not a homomorphism. The problem therefore arises to find conditions which involve the order aspect of a lattice only, not the operations g.l.b. and l.u.b., and which are necessary and sufficient for an equivalence in a lattice to be a congruence . Definition : An equivalence 7 in a poset P is directed downward, or DD, iff, for all p, q e P, p* < p and p 7 q =^ 3 q.' e P such that q ! < q and p' 7 q' . Directed upward, or DU, is defined dually. As an immediate consequence of this definition, we have: l) If 7 is DD or DU, then p| 7 f qj 7 iff there exists p 1 e p| 7 and q' e q 1 7 such that p' < q' . -5U- 2) If 7 is DU and p | 7 / q. j 7 then for all p' e p | 7 there exists a q' e q. 1 7 such that p' < q' . 3) The dual statement to 2). Theorem 9 : If the equivalence 7 in the (finite) poset P is DD or DU, then it is OP. Proof : Assume 7 is DU (DD is the dual case). Let q| 7 f p| 7 and p| 7 / ql7< Then 3p e p|/, 3 q, e q| 7 such that p < q . Since 7 is DU, ^ P P e pi 7 such that q < p . Again by DU, 3 q e q| 7 such that p < q . Continuing in this way, we get an ascending chain p < q < p < q < p < , . ., whose elements are alternatingly in p| 7 and q_j 7 „ Since P is finite, the equivalence class p| 7 is finite, hence in the above chain some element of p| 7 must eventually occur twice, say p. = p., j > i. But then p. < q. < p., hence p. - q. , hence p| 7 = q| 7, hence 7 is OP. Notice tnat the finiteness of P is essential to this proof. For infinite posets, the statement "DD => OP" is false, as can be seen from the following example. Let P be the set of all integers under their natural order, 7 the equivalence which separates even from odd integers. P is DD and DU but not OP. Definition : An equivalence 7 in a poset P has property: 1) L 2) G 3) LG iff every class of 7 has: -55- 1) a least 2) a greatest 3) a least and a greatest element. DD, DU, L and G are independent properties. An equivalence in a poset can satisfy any three of them without satisfying the fourth, as the following examples show: DD, DU, L, not G: DDj LG, not DU: The other two cases are dual to these. Theorem 10 : If an equivalence relation 7 on a lattice L is DU and G, or DD and L, then it is CF. Proof: Assume 7 is DU and G (DD and L is the dual case). Let p| 7 f r\y, p| 7 f s\y f q|7 / r|7, l\y f s\y, and let r, s be the greatest elements in their class. By the remark 2) on page $k f there exist r'., r" e r\y and b', s" e s 1 7 such that p < r 1 , p < s 1 , q < r", q < s" . Since r, s, are the greatest elements in their class, we have p a' > a and a' +b' > b' >bwe have a' + b' > a + b, hence E f C, hence E = C. The same argument with b replaced by b gives E = D, hence C = D. By duality: b 7 b =^ (a # b) 7 (a*b). Hence 7 is a congruence. Thus DD, DU and LG are necessary and sufficient conditions for an equivalence relation in a lattice to be a congruence. 1. See illustration on next page. -57- Definition : An equivalence 7 in a lattice L is a g.l.b. congruence iff for all p, p' , q e L: p 7 p' =^ (p*q) 7 (p'*q). It is an l.u.b. congruence iff: p 7 p' =^ (p + q) 7 (p' + q) . It is easily seen that Theorem 11 can be strengthen as follows: An equivalence in a lattice is: 1) a g. l.b. congruence 2) a l.u.b. congruence iff it is: 1) DD and L 2) DU and G. -58- As a conclusion, let us remark that one could define a poset P to be cross-free iff for all p, q, r, s e P: p < r, p < s, q < r, q < s implies 3 t c P such that p < t < v } q