Explicit form of operators +

L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 99 1 2 ˜ ˜ where v s 5 hsus 5 1 1j, v s 5 hsus 5 2 1j. According to the theorem of s s Section 6, it is sufficient to prove that 24–25 are necessary and sufficient conditions ˜ of a cycle in r . Let x r . . . r x r x be a cycle in r . Putting s 5 Dx , x , F 1 F F k F 1 F i i i 11 2 ˜ ˜ 1 i k, x 5 x , we derive g s 5 g Dx , x 5 1. If s [ v s holds for k 11 1 F i r i i 11 i F some s and i then the point x exceeds x in the component s. Since x 2 x 1 ? ? ? i 11 i 1 2 1 x 2 x 1 x 2 x 5 0, there exists j for which x exceeds x in the component k 21 k k 1 j j 11 1 1 2 ˜ ˜ ˜ s and, consequently, s [ v s . Analogously, s [ v s implies s [ v s for some j i j j. Thus, 24–25 obtains. The inverse statement is simple as well. n All assertions of Aizerman and Aleskerov 1983 relating to operators 0 → 5 for different 5 also can be derived on the basis of our approach. But it should be mentioned that in Aizerman and Aleskerov 1983 and Ferejohn and Fishburn 1979 results were obtained under weaker assumptions without the neutrality condition. n 9. Explicit form of operators + → 5 Using equivalent transformations, it is possible to convert any operator 10 to the form ] ] 21 21 21 21 ¯ ¯ F 9r , . . . , r , r , . . . , r , r , . . . , r , r , . . . , r 1 n 1 n 1 n 1 n 21 21 ¯ ¯ 5 F 9r , . . . , r , r , . . . , r , r , . . . , r , r , . . . , r 26 1 n 1 n 1 n 1 n 2 ˆ where F 9 is a nontrivial, monotone s.-t. operation. For example, the GLH L obtained 2 from the LH L 5 r r r by inverting of r can be presented as 1 1 2 1 2 21 21 21 21 ˆ ¯ ¯ L 5 r r r 5 r r r 5 F 9r , r , r , 1 1 2 1 1 2 1 1 2 where the operation F 9X, Y, Z 5 X Y Z is monotone and nontrivial. Applied to relations of +, the operator 26 generates only irreflexive or only reflexive relations. Let 21 us consider at first the case of irreflexive relations. Substituting r and r instead of all i i ¯ r and r in 26, we get an operator i i 21 21 Fr , . . . , r 5 Cr , . . . , r , r , . . . , r 27 1 n 1 n 1 n where the s.-t. operation C is nontrivial and monotone. The operator F is equivalent on 21 ¯ + to 26 as, first, F gives irreflexive relations and, second, xr y 5 xr y and i i xr y 5 xr y hold for x ± y, r [ +. In case 26 generates reflexive relations, we, by i i i 21 ¯ means of substitutions r and r instead of all r and r in 26, come to the operator i i i i ¯ ¯ Fr , . . . , r 5 Cr , . . . , r , r , . . . , r 28 1 n 1 n 1 n which is also equivalent on + to 26 C is nontrivial and monotone. Operators 27 and 28 are called uniform. The above procedure for obtaining them is referred to as a reduction . 2 21 ˆ ¯ For example, consider the foregoing operator L 5 r r r . The result of its 1 1 2 2 21 21 21 21 ˆ reduction is r r r 5 r . Applied to the dual operator L 5 r 1 1 2 1 1 21 ¯ ¯ ¯ ¯ ¯ r r 5 r r r , the reduction gives r r r 5 r . It easy to see 1 2 1 1 2 1 1 2 1 100 L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 s ˆ that arbitrary GLH L is reduced to r s [ h1, 2 1j where r is the first relation in the i i 21 ˆ ˆ hierarchy L r in 23, and s indicates the form of r in L r or r . The reduction of 1 i i i ˆ ¯ L leads to r or r depending on s 5 1 or s 5 2 1. i i One can restrict oneself to operators 27, 28 in order to describe all transformations n + → 5. Since all classes of 3 except for 7 contain only irreflexive relations, operators 27 are used in most cases. 21 Remark. If relations are considered up to diagonal pairs x, x then linear orders r in i ¯ 10 can be replaced by r . It transforms the operators to the form Cr , . . . , r where C i 1 n is a s.-t. operation possibly, nonmonotone. Such operators are called local Aizerman 21 and Aleskerov, 1983. We prefer to use uniform operators, as the inverse operation r ¯ employed in 27 has better properties than the complement operation r the former preserve all classes 3, the latter does not. n The following result from Sholomov 1998a allows us to describe operators + → 5 n on the basis of 0 → 5 description. Theorem. If 5 is an universally axiomatizable class and a set of operators solves the n synthesis problem for 0 → 5 then the set 9 of all uniform operators F9 obtained by n reducing operators F [ solves the synthesis problem for + → 5. Proof. Consider an operator Fr , . . . , r [ F +, 5 ; let F be in the form 27, for 1 n definiteness. Create the operator F 9r , . . . , r 5 F L r , . . . , r , . . . , L r , . . . , r 1 n 1 1 n n 1 n where L r , . . . , r is any LH that reduces to r . Let us show that F9 [ F0, 5 . i 1 n i Consider any r , . . . , r [ 0. The relation L r , . . . , r is a WO, elements x, y in 1 n i 1 n which are incomparable iff x and y are incomparable in each r , 1 j n x and y are j ¯ ¯ 9 referred to as incomparable in r if xry ∧ yrx. Let r be the WO formed from r by j j ˆ 9 9 identifying all elements incomparable in r , . . . , r , then r 5 L r , . . . , r [ +. The 1 n i i 1 n ˆ ˆ 9 9 relation r9 5 F 9r , . . . , r 5 Fr , . . . , r belongs to 5 by virtue of F [ F +, 5 1 n 1 n ˆ ˆ 9 9 and r , . . . , r [ +. As r , . . . , r can be formed by identical fissions of r , . . . , r , the 1 n 1 n 1 n relation r 5 F 9r , . . . , r is obtained from r9 by the same fission. Since 5 is 1 n universally axiomatizable, r [ 5 holds. Hence F 9 [ F 0, 5 . The set mentioned in the theorem formulation contains an operator F 0, equivalent ˆ on 0 to above F 9. The reduction of F 0 gives an uniform operator F [ 9, which is ˆ equivalent to F 9 on +. Hence, F is equivalent to the initial operator F which can be ˆ obtained by the reduction of F 9. Therefore, for any F [ F +, 5 an operator F [ 9 equivalent to F on + exists. Conversely, as F 0, 5 F +, 5 holds and the reduction leaves operators in F +, 5 , we have 9 F +, 5 . The theorem is proved. n Using the theorem and the explicit form of operators 0 → 5 Section 8, and taking L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 101 s into account that the reduction of a GLH gives r , we obtain the following results about i n operators + → 5 perhaps, most of them are known. a a F [ F +, 5 , 5 [ h+, 0, 6, j iff F is equivalent to an operator r , a [ h1, i 1 2 1 j to an operator r if F [ F +, 5 . For F [ F+, + the result is in Monjardet i 1978, for instance. a s b F [ F +, 3 iff F is equivalent to an operator r , a [ h1, 2 1j to an 1s k i s s 1 operator r if F [ F +, 3 . 1s k i s a s c F [ F +, 7 iff F is equivalent to an operator in the form r or 1s k i ] s ] a 21 1 s r , a [ h1, 2 1j r or r if F [ F +, 7 . 1s k i s 1s k i 1s k i s s s a d F [ F +, iff F is equivalent to an operator r G, a [ h1, 2 1j, where G is i 1 1 any operator in form 27 to an operator r G where G is any monotone operator i 1 1 in form 27, if F [ F +, . For F [ F +, the result is in Blau and Deb 1977. n

10. Explicit form of operators 4