LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 5I0.&4 no. 343 -34 & coip. 2 The person charging t^^^XarV^rom Tatest Date stamped below. ^^^ ^^^^^^^ The.., n,o.na,ion, and -— ^-"^J^,, i„ dismissal from for disciplinary aC.on and may the University. r»nfer 333-8400 TO renew .ail Telephone Center ^^^bANA-CHAMPA^ UmVERS.TY O^OHNO^^""'^''^ == Li6l_O-1096 'T jLls.V Report No. 3^8 August 12, 1969 yn^LZt CONSISTENT PROPERTIES OF COMPOSITE FORMATION UNDER A BINARY RELATION by John C. Schwebel Bruce H. McCormick COO-1018-1189 Hi m \\\L DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS 5fP 24 Digitized by the Internet Archive in 2013 http://archive.org/details/consistentproper348schw COO-1018- 1189 Report No. 3^8 jerson charging this material is re- ible for its return on or before the t Date stamped below. eft, muHlation, and underlining of books e reasons for disciplinary action and may suit in dismissal from the University. University of Illinois Library CONSISTENT PROPERTIES OF COMPOSITE FORMATION UNDER A BINARY RELATION* John C. Schwebel Bruce H. McCormick August 12, 1969 )artment of Computer Science University of Illinois Urbana, Illinois 618OI ^Supported by Contract AT(11-1)-1018 with the U.S. Atomic Energy Commission. INTRODUCTION In this paper ve study the properties of a relation, H, between tvo sets, X and Y, H ^ X x Y. We wish to determine when the existence of H between some elements of X and Y implies the existence of H between other composite elements of X and Y. The aim is to characterize a rela- tion by its extension or restriction to composite elements. Composite elements are formed by containment relationships among other elements. Thus the properties studied depend on the struc- ture on the X and Y sets. We first consider the general case of latticesj M and N, on X and Y respectively, where the containment relationships pro- vide the partial ordering on the sets. Here the meet and join operations are available. Later, Boolean algebras and the additional structure implied by the complementation operation are considered. Systems composed of lattices M, N and relation H are first formally defined. Here we define a group of system transformations to simplify later proofs. The properties of systems are then defined. All possible con- sistent combinations of property values are established by the use of reduction theorems and by the construction of example systems. Using thirteen independent properties there are exactly one hundred and forty seven possible combinations of property values. This set can be generated from a minimum set of forty values by the group of system transformations. These results are quite general and may have applications to areas requiring a class if icatory or composite analysis approach. Some of the thirteen independent properties defined in this paper are related to other concepts of structure mappings. The preserva- tion of corresponding relations of the same rank on two sets, which includes three of the thirteen independent properties and two logical combinations -1- of other properties , may be represented by a generalized pair algebra as defined by Yeh [l]. The preservation properties are also generalizations of various types of functional homomorphisms and generalized congruence relations, which are explained in Yeh [2] and Liu [3]. Motivation The current work is intended as a theoretical foundation for the study of graphical transformations on a set of objects interrelated by binary relations. The properties defined here arose in attempting to characterize binary relations of image processing. In this case, elements of a directed graph typically correspond to "regions" of the image, and composites (unions of elements) correspond to figures of the scene. The properties specify ways in which relations extend to unions. After characterizing a set of relations, it may be possible to determine which graph transformations are valid and desirable in order to form composites from elements, operations and relations. A paper in progress pursues this possibility by enumerating and considering some specific graph transformations. Transformations which are implied by properties of a single relation as defined in this paper are exhibited. Relations of a transformation are next considered jointly, and the partitions induced on a class of relations by their joint behavior in the transformation are studied. -2- LATTICE PRELIMINARIES In the following assume M and N are lattices on X and Y with the defining operations meet,n , and join,U , M = (X,U ,n), N = (Y,U , n). The basic principles of lattices used in this paper may "be found in Chapter one of Birkhoff [k] or in Chapters one and two of Szasz [5] . Lattice Duality The dual of a lattice theoretical proposition inyolying the operational symbols H and U is obtained by interchanging H and U every- where in the proposition. Denote the dual of a proposition P by D(P). By the lattice duality principle P is true if and only if D(P) is tri^a, i.e. , P^D(P). Given a lattice M = (X, U fl ) ve can define a new lattice M^ = (X, U^, n^) by a n^ b = a U^ b a U b = a n b for all a, b e X. The lattice duality principle shows that M is a lattice. M is called the dual of M.^ , M = D(]yL ) . The ordering of M is reversed from that of M . We use less than or equal, <_, and greater than or equal, ^,to represent the lattice ordering. Since a <. b. b^ a n b = b D(a < b) = a > b. ■3- Clearly D(D(M)) = M. If D(M) is lattice isomorphic to M then we say M is self-dual . A lattice isomorphism is a one-to-one and onto mapping which preserves both lattice operations. We will represent finite lattices by diagrams. The upward direc- tion in the diagram corresponds to "greater than". If point a is higher than point b in the diagram, and if points a and b are connected by a branch, then a > b. The diagram of the dual lattice may be obtained by inverting the diagram of the lattice. -h- 3. DEFINITION OF A SYSTEM We consider a class of systems^^t^ where S *y^ is a system lujji. Let X, Y be arbitrary sets, H a binary relation from X to Y, HcX x Y, and M and N be lattices on the sets X and Y respectively, M = (X,U , n ), N = (Y,U , n ) . Then S «j^ if and only if S = (H, M, N) for some X, Y, H, M and N. Group of Operators on a System Given a system S b a H b^ £. bg > b^ =^ a Hbg ^ a ->■ > b PI : Propagates Inverse Images Up a -> b a^ Hb g. a^ 1 a^ s^ a^Hb ^ > a ->■ b P2: Propagates Images Down a ->■ b aHb e- b <_ b ^ aHb ^ ^2_ -^ -2 - 1^ 2 a ^ < b P2 : Propagates Inverse Images Down a -> b a^Hb C-a^ 1 a^^ a^Eb >| < a ->■ b P3: Joins Images a ^ b aHb £. aHbg^^a H(b U b ) a ^ b a -> b^ U b^ P3 : Joins Inverse Images a ->■ b a Hb g. a Hb =^ ( a U a ) H b a ->■ b ^ a U ap -> b PU: Meets Images a -> b aJTb^ fil aHb2=^aH(b^ H b^) a -> b^ a ^ b^ n b^ -10- PU : Meets Inverse Images a -^ h a Eh t a Kb =^(a H a )H1d a -> b a^Hb^ %. a^Hb^ ^{a^ U a^) H (b^ U b^) i a n a -^b P5: Preserves Joins a -> b a^-b^ a^ U a^ -> b^ U tg P6: Dualizes Joins a^ ->■ b a^Hb^ S- a^Hbg^^ (a^ U a2)H(b^ D b^) a^ -^ b^ a^ U a^ -> b^ n bg p6 Dualizes Meets a -^ b^ a^Hb^fi. a^Eh^^ia^ fl a2)H(b^ U b^) a^ -> b^ a^ n a^ -»- b^ U b^ P7 : Preserves Meets a ^ b a^Hb^ Sr a^Hb^^ (a^ n a^) H (b^ n b^) a^ -> b^ a^ n a^ ^ b^n b^ -11- other properties can be defined which are logical combinations of the twelve independent properties. We show four such properties below. P2 6) P2 : Preserves Less Than a -> b PI E- PI : Preserves Greater Than P2~ S- PI: Dualizes Less Than < a -> < b a -> b f > a ->■ > _ b a -> b i' < a -*- > b PI £- P2: Dualizes Greater Than a -> t 4 > a -> < b -12- Permutations of Property Valiaes Tatle k shows T(P) for all twelve properties P and all T in G. Since each T(P) is a permutation of the properties P, we can deter- mine the true or false values of the properties T(P) of the system T(S), given the properties P of S. Table h also shows that the twelve properties are partioned into three sets of four properties such that each set is closed under all T in G. These three sets are separated by double lines in Table U, and will be referred to in the following as the PI set, the P3 set, and the P5 set. Any combination of values of the twelve properties will be referred to by an index, (i, j, k), where i, j, k are indices of the values of the properties in the PI, P3, and P5 set respectively. The values of the indices i, j, k are determined by partioning the sixteen possible true or false values of four properties into the following six sets, with indices 0, 1, 2, ..., 3' Index Set 0000 1 0001, 0010, 0100, 1000 2 1010, 0101 3 0011, 0110, 1001, 1100 k 1110, 1101, 1011, 0111 5 1111 Each of these six sets of property values is closed under all T in G, and each element of each set is a generator of the set vinder T in G. Thus the index, (i, j, k) of any value combination of the twelve properties does not change imder T in G. That is, if values P are indexed by (i, j, k), the values T(p) are also indexed by (i, j, k). These facts will allow us to easily derive a minimum set of generators for all con- sistent cases of properties. This derivation will be carried out after proving theorems interrelating the twelve properties in the next section. -13- T(P). \ p X 1 1-1 2 2-1 3 3-^ U u-1 5 6 T 6-1 D 2 2-1 1 1-1 i| 4-1 3 3-1 T 6-1 5 6 \ 1 2-1 2 1-1 3 u-1 h 3-1 6-1 T 6 5 \ 2 1-1 1 2-1 1+ 3-^ 3 1.-1 6 5 6-1 T I 1-^ 1 2-1 2 3-^ 3 u-1 k 5 6-1 7 6 ID 2-1 2 1-1 1 h-"- k 3-1 3 T 6 5 6-1 ^\ 1-1 2 2-1 1 3-^ k k-^ 3 6 T 6-1 5 ^=B 2-1 1 1-1 2 k-^ 3 3-^ ^ 6-1 5 6 7 TABLE U -14- 5. ENUMERATION OF POSSIBLE COMBINATIONS OF PROPERTIES We now wish to determine all logically consistent combinations of the twelve properties P of a relation H in a system S = (H, M, N). This section reduces the number of possibly consistent value combinations by proving and applying theorems which interrelate the twelve properties. The next section proves that this set of theorems is complete by constructing examples of systems having all value combinations in the reduced set . We will prove theorems involving properties P and expressions of logic. Corresponding to a theorem. A, are eight theorems T(A) , T e G, which may or may not be independent of A, which are obtained by replacing P by T(p) for all P involved in A. Since T(P)^^P, and the properties involve arbitrary system S, it follows directly that T(A)^^A. That is, a proof of A implies a proof of T(A). This will considerably simplify the proofs to establish all consistent cases. Theorems Relating Properties In Table 5 ve list twenty independent theorems, numbered one through twenty, and a diagrammatic representation of the theorems in a form which will make the application of the theorems in the reduction process readily apparent. -15- Theorem Dlagrsm 1 1-1 2 2-1 3 3-1 u ri 5 6 T 6-1 1. 1^ 3 2. 1-1^3"^ 1^ 1 t.«^ r^^ 3. 2^1+ 1+. 2"^=^U"^ ■t^ ■■E>- -^*" 5. 5=^3 ■•^ 6. 5^3-^ 7. 6^3"^ ^^ 8. 6=^1+ 9. l^h <1 10. 7-^^"''' 11. 6-^3 - P:>^ "l:^ r.^^ "i^^ i-^ U*" 1 [>• 1 r** TABLE 5 -16- Table 6 shows that transformations , T, applied to theorems one, five and thirteen generate all twenty theorems, and that these twenty theorems are closed "under T in G. T(A) \* \ 1 5 13 ^L 1 11 11+ I 2 6 15 ^L 2 7 16 ^R 3 8 IT D 3 9 18 ^R h 12 19 ID h 10 20 TABLE 6 Below we give proofs of theorems one, five and thirteen, thus implying proofs of all twenty theorems. Thm. 1 PI ^P3 Proof: a H b 9. aHb (by PI and b^ U b^ ^ b or b U "b > b ) a H (b^ U bg) Q.E.D. Thm. 5 P5 =^P3 Proof: a H b^ aHb^ =^ (by P5) (aua) H (b^Ub^) =^ a H (b^ U b^) Q.E.D. -IT- Thm. 13 PI 6-P3 "'■=^P5 Proof: a^Hb^ g: a^Hb^ ^ (by (t^Ub^) ^ b^ and (b^Ub^) ^ b^ and Pi) \ H (b^Ubg) S-a^ H(b^Ub2) ^(by PS"^) (a^Ua^) H (b^Ub^) Q.E.D. Figure 1 represents all of the twenty theorems by an implication graph whose nodes represent the twelve properties. The following symmetries of the properties and implications are apparent from Figure 1. The transformation D corresponds to reflection about a line -1 -1 ^ through nodes 1 and 2 . D corresponds to reflection about a line Li through nodes 3 and h. D(P) corresponds to reflection through the center of the diagram on a line through node P, for any property P. Reduction of Possible Consistent Cases 12 There are 2 = i+096 possible combinations of binary values of the twelve properties. By applying the twenty theorems, the number of possible consistent combinations is reduced to 136. Further the set of values, y, where y= {v}, is partitioned into subsets, called blocks, by the equivalence relation: v is equivalent to v' if and only if v = T(v' ) for some T in G. Every element in each block is a generator of the block under T in G. We will show that there are 33 consistent blocks and, there- fore, 33 generators can generate all consistent cases. The reduction process and derivation of a minimum set of genera- tors is illustrated in Table 7- Each combination is indexed by (i,j,k), as explained in the previous section. We will refer to a value, v, by v v^ v_, where v, is the h bit value of the PI set of properties, v of the P3 set and V of the P5 set. In the derivation, for each J = 0, 1, 2, 3, ^ all possible consistent values of v and of v are listed for one possible -18- SYMMETRY SYMMETRY FIGURE I -19- value of V . For j = 5) since v^ has only one value* all possible con- sistent values of v„ are listed for one value of v^ for each i = 1, 2, 3, ^. For i = 0, j = 5> one value of v is listed for each k=0, 1, 2, 3, ^, 5. It is easily seen that the values in Table 7 are generators for all possible consistent values. For any consistent value v v v , with index (i,j,k), we can always find a T such that T(v v v ) = v' , v', v', and v' , v', v' is in Table 7^ There are three cases: 1) j=0, 1, 2, 3, U and T is the transformation which takes V into the v' used in the table. 2) j=5ji=lj2, 3, ^ and T takes v. into the v' used in the table. 3) j=5*i=0 and T takes v into the v' used in the table. Since there are 32 different (i,j,k) blocks in Table 7» and any T does not alter the block of its operand, there must be at least 32 consistent blocks. By applying transformations to the values in Table 7» we see that there are at most 33 consistent blocks. The set indexed by (l,i|,3) requires two generators, while the other consistent (i,j,k) sets require only one generator. We index the 33 possibly consistent blocks by (i,o,k) if (l,j,k) ^ (1,J4,3), and by (l,U,3) (l) and (l,U,3) (2). Table 8 lists a set of 33 generators from the values of Table 7- The values in Table 7 which are not used in Table 8 have an entry under column T specifying the transformation which generates them from the value in Table 8 in the same block. Table 8 also lists the number of elements in each block. All 136 values are listed in Appendix 1. -20- TAJ,.LE 7 Reduction of Set of Possible Consistent Values of Twelve Properties 1. All possible values in all i,k sets are listed for one rep- resentative value in the j set, for j = 0,1,2,3,^. Index T Properties i j k 1 1'^ 2 2"^ 3 3"-^ h k~^ 5 6 7 6-^ 000 0000 0000 0000 010 0000 0001 0000 110 0001 0000 020 0000 1010 0000 120 0010 . 0000 120 D 1000 0000 220 1010 0000 030 0000 0011 0000 031 0000 0010 131 0001 0010 131 I 0010 0010 331 0011 0010 0000 oUo 0000 1110 OUl 0000 0100 Oi+1 I-E 0000 1000 Ol+3 0000 1100 lUl 0010 0100 1U3(1) 0010 1100 lJ+3(2) 0100 1100 3U3 Olio 1100 lUl \ 1000 1000 143(1) ^R 1000 1100 2U3 1010 1100 3i+3 °E 1100 1100 Ui^3 1110 1100 -21- 2. All possible values for all k sets are listed for j = 5 and for one representative value in the i set, for i = 0,1,2,3,U,5. Index T Properties i j k 1 1~^ 2 2"^ 3 3"^ h k~^ 5676-^ 050 0000 1111 0000 051 0001 052 j 1010 053 0011 05U 1110 055 1111 153 1 0001 1111 0011 I5U ! 0111 15ii 1 \ 1011 155 1 1111 255 j 1010 1111 1111 355 U_55 555 0011 1110 1111 1111 1111 1111 0111 1111 1111 1111 -22- TABLE 8 Minimum Set of 33 Generators of 136 Cases Block Index i j k 1 1-1 2 2-1 3 3-^ k k-^ 5676-1 Order of Block 000 0000 0000 0000 1 010 0000 0001 0000 h 020 0000 1010 0000 2 030 0000 0011 0000 k 031 0000 0011 0010 k oUo 0000 1110 0000 k ■ Ol+l 0000 1110 0100 8 0U3 0000 1110 1100 k 050 0000 1111 0000 1 051 0000 1111 0001 k 052 0000 1111 1010 2 053 0000 1111 0011 k 05U 0000 1111 1110 h 055 0000 nil nil 1 110 0001 0001 0000 h 120 0010 1010 0000 k 131 0001 0011 0010 8 ll+l 0010 1110 0100 8 1H3(1) 0010 1110 1100 8 li+3(2) 0100 1110 1100 k 153 0001 nil 0011 k 15h 0001 nil 0111 8 155 0001 nil nil k 220 1010 1010 0000 2 2h3 1010 1110 1100 k 255 1010 nil nil 2 331 0011 0011 0010 k 3h3 Olio 1110 1100 8 354 0011 nil 0111 k 355 0011 nil nil k UU3 1110 1110 1100 h ^55 1110 nil nil k 555 1111 nil nil 1 -23- 6. CONSTRUCTION OF EXAMPLE SYSTEMS In this section an example system is constructed for one repre- sentative in each of the 33 possible consistent hlocks, thus proving that there are exactly 136 possible consistent combinations of values of the twelve properties. Factoring of Property Set In order to determine a procedure for explicitly constructing examples of systems with given properties, we wish to determine the largest sets of properties which are completely independent of each other. Thus, we wish to factor the properties into sets whose values may be assigned independently, and which have a maximum of internal constraints so as to give a minimum number of values within each set. Figure 1 shows that at least two nodes must be removed to obtain independent sets. By removing 6 and 6 , the ten remaining properties factor into two sets of five properties, 1, 1 , 3, 3 ,5 and 2, 2 , ^, h , Ti such that the two sets are congruent under duals. That is, the properties and theorems of one set are duals of the other set. Each set is aJ-So closed under inverses. Thus, each set has the same possible value assignments, and the two sets combined have all possible products of assignments of each set. This is an optimal factoring and the one that will be considered below. An exactly analogous situation occurs if 5 and 7 are removed. The next best factorings are obtained by removing 7 and 6 , or 5 and 6, or 5 and 6 , or 7 and 6, which result in independent sets of 2 and 8 elements symmetric under D„ or D^' n L Now we will construct examples for each dual set of five properties, then determine a procedure to construct examples for ten properties from examples of each set of five, and finally alter the application of the procedure to obtain examples when permissible values of the other two properties, 6 and 6 , are also considered. _2l|- Simplest Examples for Factored Sets The theorems eliminate all but ten com,binations of the five properties 1,1 ,3,3 ,5 and the same ten combinations of 2, 2 , U, 1+ ,7- These ten possibilities are given in Table 9 below. To prove that all ten cases are consistent, diagrams of systems satis- fying each case are shown in Figure 2. The example corresponding to each case is designated at the right of entry in Table 9« The Dual examples correspond to the dual properties. S, is the null example, i.e. S^ = D(S^) = (0, 0, 0). 1 1-1 3 3-1 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Example KSg) 1(63) KSg) TABLE 9 The examples given use the simplest lattices and relations pos- sible. That is, if we define a system S = (H, M, N) of a set;^' to be simplest if 1) both M and N have the fewest points and the most complete ordering of any S inj^' , and 2) H has the fewest elements of any S in^' , then we can show that each of our examples is simplest in the set^' of S in ;^ satisfying the given properties. -25- D(S,) D(S3)=S3 D(S4) D(S5) = S5 D(S6) D(S7) = S7 FIGURE 2 To show that these examples are simplest we need only show that S^ , S , ... S are simplest. Below are diagrams of all lattices of less than or equal to four points, ordered according to strictly decreasing simplicity. Let L. > L, mean L. is simpler than L.. J 1 J L^ > L^ > L3 > L^ > L^ > Lq I . ^ Let (Hj be the number of elements in H. Table 10 shows M, |h| and N for the examples. s M H N ^1 ^0 • ^0 ^2 h ■ 1 • \ S ^5 • 2 ■ h h h ■ 1 ' \ ^5 ^2 ' 2 ■ \ h s ■■ 2 ' \ 'l s ■■ 3 • S TABLE 10 -27- The following facts may be easily established by enimierating all simpler cases: 1. -I (Pl"^) =^ M > L 2. -1 (PI ) ^ N > L^ 3 -1 (P3"^) ^M > Lj^ 4. -I (P3) =^ N > L^ 5 . — I ( PI or Pl""^ ) =^ I H I ^ 1 6. — t (P3 or P5 or P3~"'") ^ IhI > 2 These facts and Table 10 show directly that S, , S^, ... , S^ are Id b simplest. Additionally we need only show that |h| = 2 is not possible for S . Again this can be seen by enumeration of cases. Thus, S , S ,..., S are simplest examples. Combinations of Ten Properties Next consider the ten properties 1,1 , 3, 3 ,5 and 2, 2 , i+ , U ,7 together. There are one hiondred possible consistent combinations of these ten properties, obtained by taking all combinations of the ten consistent cases for each set of five properties. We can prove that all one hundred cases are consistent by giving a procedure to construct an example satisfying the ten properties from two simplest examples satisfying the two sets of five properties. This procedure is given below. Let S = (H , M , N ) be the example satisfying a combination -1 -1 of properties 1, 1 , 3, 3 , 5 and S = (H , M , N ) be the example -1 -1 d d d for 2, 2 , i+, U , 7. Construct a new example S = (H, M, N) as follows: \ I -28- 1. Form M (N) by joining the minimum element of R (N ) to the maximum element of M (W ). 2. Form H' in X x Y by adding elements to H, implied by the 2, 2 , ii , i+ , T properties of S . Form H' in X X Yp by adding elements to H implied by the 1, 1 , 3j 3 , 5 properties of S . 3. Take H = H ' U H'. Then we have a lattice M on X, N on Y, and H c X x Y. X is a set with distinct points corresponding to X - min(M.^ ) and X - max(M ) and one point corresponding to the two points min(M ) , m8Lx(M ). Y is obtained similiarly. For example if IVL , M , N , N all have the structiire: then S = (H, M, N) has the structure; «1 «2 -29- Thm. If S and S are simplest examples with properties 1, 1"^, 3, 3"^, 5 and 2, 2"^, h, k~^ , J respectively, then S as constructed above has the same 1,1 ,3, 3 ,5, 2, 2 i h, k ,7 properties. Proof: i) H ' _c: X x Y ; since H ^ X x Y and if H' has an element (x', y) with x' in (X-X ), then JX | > 1 and S must satisfy PI , since PI is the only property that can move an element of H above the max(M ) . But all simplest examples with PI =1 have |X | = 1. Similiarly H' can have no element (x' , y') with y' in (Y-Yg). Thus H' ^ X x Y , or the properties of S can move no elements of H upward into (X-X^) X (Y-Yg). ii) In forming H' , the properties of S can only move elements of H downward. iii) Since no higher elements are added in (X-X )x (Y-Y ), S fails to satisfy the same properties as S in a simplest example. iv) Adding lower elements in a simplest example cannot change any of the S properties. Thus S has the same properties as S • The dual argument shows S has the same properties as S . Thus S has the same 1 , 1~ , 3, 3~ , 5 properties as S and the same 2, 2""^, h , k~ ,7 properties as S . Q.E.D. This procedure will not necessarily give the simplest example in each case. It does establish that all one hundred values are consistent. -30- Consistent Values of Twelve Properties We will now construct 33 examples, one for each block, of systems considering all twelve properties. The procedure given for constructing example systems having a desired value of ten properties is used and then modified if possible to give the proper value of properties 6 and 6 First we notice that we can reduce the set of 100 combinations of ten properties to a smaller generator set by using the transformations I, D, and ID, under which the 10 properties are closed. Denote the 100 possible values by (x,y) where x and y gives the index of the value of 5 properties in Table 9- Then, the set is reduced to 3^ values, indicated by O's or I's in the matrix of Table 11, by using the four transformations below: 1) D(x,y) = (y,x), so only the lower triangular of the ten by ten matrix is necessary. 2) I(l(x), I(y)) = (x,y) so columns l(2), l(3), l(6) are not needed. 3) I(l(x),y) = (x,y) for y = 1, U, 5, 7 so that these four y columns are not needed with the last 3 rows . h) ID (l(x),y) = (l(y), x) so only the lower triangular of the submatrix consisting of rows l(2), l(3), l(6) and columns 2,3,6 is necessary. Next we consider the possible values of P6 and P6 consistent with the theorems for each of the 3^ cases. This adds only l6 values. We then pick one combination for each of the 33 blocks. Table 12 shows the 33 representatives. To the left of the value are the x and y indices. Only the values of x and y marked by I's in Table 11 are used. To the right of the entry in Table 12 is the block index, and columns P and S. -31- A modified procedure is used to construct example systems for each of the 33 values. These examples are displayed in Figure 3 by diagrams, where M is represented by the left lattice, N by the right lattice, and H by directed branches from left to right. The modified procedure used first constructs the example for the ten properties as explained above, then modifies it intuitively, either to satisfy the values of p6 and P6 , or to simplify the example, or both. Properties of the final example arrived at in Figure 3 are indicated by coliomns P and S of Table 12. Column P is marked if the example is exactly the same as the example produced by the original procedure, and column S is marked if the example is simpler. Example systems for all 136 possible values can be obtained by applying a proper transformation, such as the one listed in Appendix 1, to the representative in Figure 3. 1 2 3 k 5 6 7 1 1 2 1 1 3 1 1 1 k 1 1 1 5 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1(2) 1(3) 1(6) 1 TABLE 11 -32- Block # X y , 1 1-1 2 2-1 3 3-1 k U-1 5676-1 Index i j k p s 1 1 1 1111 1111 nil 555 1 2 2 1 1011 1111 nil i^55 1 3 3 1 1011 1011 0011 kk3 1 k h 1 0011 1111 nil 355 1 5 5 1 0011 1111 0111 35U 1 6 6 1 0011 1011 0011 3U3 1 T 7 1 0011 0011 0010 331 1 8 2 2 1010 1111 nil 255 1 9 3 2 1010 1011 0011 21+3 1 10 U 2 0010 1111 nil 155 1 11 1+ 2 0010 1111 1110 15i+ 12 5 2 0010 1111 Olio 153 13 6 2 0010 1011 0011 li+3(l) 1 Ik 6 2 0010 1011 0010 li+1 15 1(6) 2 0010 0111 Olio li^3(2) 1 16 7 2 0010 0011 0010 131 1 17 3 3 1010 1010 0000 220 1 18 6 3 0010 1010 0000 120 1 19 7 3 0010 0010 0000 no 1 20 ii 1+ 0000 1111 1010 052 21 1+ k 0000 1111 1011 05U 22 k h 0000 1111 nil 055 1 23 5 h 0000 1111 0010 051 2U 5 k 0000 1111 0011 053 25 6 h 0000 1011 0010 Oi+1 26 6 k 0000 1011 0011 0U3 1 27 7 k 0000 0011 0010 031 1 28 5 5 0000 1111 0000 050 29 6 5 0000 1011 0000 oi+o 30 7 5 0000 0011 0000 030 1 31 6 6 0000 1010 0000 020 1 32 7 6 0000 0010 0000 010 1 33 7 1 7 0000 0000 0000 000 1 1 TABLE 12 -33- ^ 7 8. 10. 12, FIGURE 3 SHEET 1 13. 14. 15. 16. 17 18. 19. 20. 21. 22. 23. 24. FIGURE 3 SHEET 2 -35- 25. 26 27. 28. 29. 30. 31 32. 33. FIGURE 3 SHEET 3 -36- 7 . COMPLEMENTATION We now introduce one additional property for the case when M and N are Boolean algehras. Then, a unique complement, x, exists for each element x in M or N. We will use DeMorgan's laws: (a U ^) = a n b (a n b) = a U b and the identity, a <_ b^a >_ b, which hold in a Boolean algebra. The new property is P8: Preserves Complements a H b =^a H b P8 is invariant under T, i.e. T(p8) = P8 for all T in G. Thus, T still permutes the set of all thirteen properties. The following theorems relate P8 to the first twelve properties. Thm. 21 Proof: P tP8=>D(P) Because of DeMorgan's laws, D(P) can be obtained, if P8. is true, by complementing variables on the left hand side of the implication sign in P, applying P to the complemented variables, and then complementing the elements of H in the resulting statement. Q.E.D. For example we prove : PI g. P8 ^ P2 . Proof: a H b^ g.b2 <.b^^(by P8) I H 1)^ g. b^ ^ b^ ^ (by PI) a H b. a H b. a H b. (by P8) Q.E.D. -37- Thm. 22 PI ^ PI ^ g. P2 g. P2 """-^^ P8 Proof: Let represent the minimum element of the Boolean algebra. a H b =^(by P2 €- P2~^) H ^(by PI fi. Pl"^) a H b Q.E.D. Now we wish to determine all possible consistent cases of all thirteen properties when M and N are Boolean algebras. Applying Theorem 21 we get : P8=^(P1 = P2) g. (Pl~-^ = P2~^) e. (P3 = Pi+) S- (PS"""" = Pi^"''') S- (P5 = PT) VI (p6 = p6"^). Thus P8 cannot be true unless all properties imply their dual properties. This could also be proven by showing that Boolean algebras are self-dual lattices with complementation as the isomorphism between L and D(L) . The theorems determine P8 uniquely in all but eleven of the one hundred thirty six cases of the first twelve properties, or seven of the thirty three generator cases, giving a possible one hundred forty seven total cases or forty generator cases . We show that all of these cases are possible by giving examples of the fourteen generators where P8 is not uniquely determined by the first twelve properties. The diagrams of Boolean algebras in Figure h are n-cubes where diagonal points are complements. Table 13 gives the index of the example in Figure k or Figure 3 which corresponds to each of the fourteen cases. Thus , examples of all one hundred and forty seven cases of the thirteen properties can be generated from the examples in Figures 3 and h. -38- Index of Basic Case Illustration of First 12 Properties p8 Nimber 8 CI 8 1 C2 17 C3 IT 1 17 20 Cl| 20 1 20 22 C5 22 1 C6 28 C7 28 1 28. 31 C8 31 1 09 33 33 33 1 CIO TABLE 13 -39- CI C2. C3. C4. C5. C6. C7. C8. C9, CIO. FIGURE 4 REFERENCES 1. Yeh, Raymond T. , "Generalized Pair Algebra With Applications To Automata Theory", JACM 15, 2 (April 1968), 30l+-3l6. 2. Yeh, Raymond T. , "Structural Equivalence of Automat a", Paper pre- sented at the 9th Switching and Automata Theory Symposium. 3. Liu, C.L. , "Lattice Functions, Pair Algebras and Finite-State Machines", JACM, Vol. l6. No. 3, July I969, pp. UI+2-U5U. k. Birkhoff, G. , "Lattice Theory", American Mathematical Society, 1963, 277 pages. 5. Szasz, Gabor, "introduction to Lattice Theory", Academic Press, New York, I963, 229 pages. -Ul- APPENDIX 1 All 136 Consistent Cases are listed by block. The first entry in each block is the generator in Table 8. A transformation which generates each of the other elements from the first is specified under Coliimn T. Block Index i .1 k 11^2 2""^ 3 3"^ h h~^ 5616 -1 000 0000 0000 0000 010 020 030 031 oUo 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0010 0000 0100 0000 1000 0000 1010 0000 0101 0000 0011 0000 Olio 0000 1001 0000 1100 0000 0011 0010 Olio 0100 1001 0001 1100 1000 1110 0000 1101 0000 1011 0000 0111 0000 I D ID R D I D ID A-1 -U2- i J k 04l 0U3 1 1-1 2 2-1 3 3-^ k k-^ 0000 1110 0000 1110 0000 1101 0000 1101 0000 1011 0000 1011 0000 0111 0000 0111 0000 1110 0000 1101 0000 1011 0000 0111 ^676 0100 1000 0001 1000 0001 0010 0100 0010 1100 1001 0011 Olio -1 I ID. D ID ID, I D ID R 050 051 052 053 05U 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0000 1111 0001 1111 0010 1111 0100 1111 1000 1111 1010 1111 0101 1111 0011 1111 Olio 1111 1001 1111 1100 1111 1110 1111 1101 1111 1011 1111 0111 D D, I ID D D D, 055 0000 1111 1111 A-2 i j k 110 120 131 lUl li^3(l) 1 1-1 2 2-1 3 3-^ k k-^ 0001 0001 0010 0010 0100 0100 1000 1000 0010 1010 0001 0101 1000 1010 0100 0101 0001 0011 0001 1001 0010 0011 0010 Olio 0100 1100 0100 Olio 1000 1100 1000 1001 0010 1110 1000 1011 0010 1011 1000 1110 0001 1101 0100 0111 0001 0111 0100 1101 0010 1110 1000 1011 0010 1011 1000 1110 0001 1101 0100 0111 0001 0111 0100 1101 A- 3 -kh- 5 Gi e 0000 0000 0000 0000 0000 0000 0000 0000 0010 0001 0010 0100 1000 0100 1000 0001 0100 0001 0010 1000 0001 0100 0010 1000 1100 0011 0011 1100 1001 Olio Olio 1001 -1 I D ID I D ID I D \ ID D I ID ID R D I ID ID R i j k 1U3(2) 153 151+ 155 220 2U3 255 1 1-1 2 2-1 3 3-1 k U-1 5 6 7 6- 0100 1110 1100 1000 1101 1001 0001 1011 0011 0010 0111 Olio 0001 1111 0011 0010 1111 Olio 0100 1111 1100 1000 1111 1001 0001 1111 0111 0001 1111 1011 0010 1111 0111 0010 nil 1110 0100 1111 1101 0100 1111 1110 1000 1111 1101 1000 1111 1011 0001 nil nil 0010 nil nil 0100 nil nil 1000 nil nil 1010 1010 0000 0101 0101 0000 1010 1110 1100 1010 1011 0011 0101 1101 1001 0101 0111 0110 1010 nil nil 0101 nil nil -1 I D ID I D ID R D ID ID, I D ID D I ID i J k 331 3i+3 33h 355 UU3 U55 1 1-1 2 2-1 3 3"-^ h h~^ 5 6 7 6- 0011 0011 0010 1100 1100 1000 Olio Olio 0100 1001 1001 0001 Olio 1110 1100 1001 1011 0011 0011 1011 0011 1100 1110 1100 1001 1101 1001 Olio 0111 Olio 0011 0111 Olio 1100 1101 1001 0011 1111 0111 1100 1111 1101 Olio 1111 1110 1001 1111 1011 0011 1111 1111 Olio 1111 1111 1001 1111 1111 1100 1111 1111 1110 1110 1100 1101 1101 1001 1011 1011 0011 0111 0111 Olio 1110 1111 1111 1101 1111 1111 1011 1111 1111 0111 1111 1111 -1 D D I ID ID R D ID. 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