Explicit form of operators 0

L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 97 n Proof of Theorem. Let F be an operator 0 → 5 for an universally axiomatizable class 5. Consider some axiom 4 used in definition of 5. Let x , . . . , x be any points 1 s n in R . Let us take both relations r , . . . , r [ 0 and elements x ,.., x [ A guaranteed 1 n 1 s by a of the Lemma. The relation r 5 Fr , . . . , r of 5 satisfies the formula Px , . . . , 1 n 1 x in 4. According to a, x r x ⇔ x rx and, therefore, Px , . . . , x is true for r . s i F j i j 1 s F Since points x , . . . , x are arbitrary, r meets the axiom 4. As this reasoning can be 1 s F applied to any axiom used in the definition of 5, it gives r [ 5. F Conversely, let us show that r [ 5 implies r 5 Fr , . . . , r [ 5 for any r , . . . , F 1 n 1 n r [ 0. Consider an axiom 4 and any x ,.., x [ A. Let x , . . . , x be points of R n 1 s 1 s guaranteed by b of the Lemma. Because r meets the formula Px , . . . , x , the F 1 s relation r satisfied Px , . . . , x . To complete the proof, it is sufficient to remark about 1 s arbitrariness of both considered axiom and elements x ,.., x . 1 s n As the result, the synthesis problem for operators 0 → 5 is reduced to the description problem for ordinal relations of 5 studied in Section 6. n

8. Explicit form of operators 0

→ 5 The aggregation operator, associated with a lexicography l, is called a linear hierarchy LH and denoted by L. By 13, its representation 9 is L 5 r ∨ r r ∨ ? ? ? ∨ r r ? ? ? r r . 23 1 1 2 1 2 n 21 n ˆ Analogously, a partial LH PLH L9, a generalized LH GLH L, and a generalized ˆ ˆ ˆ PLH GPLH L 9 are defined using suitable lexicographies l9, l and l 9. The GLH is 21 formed from LH by replacement of some relations r by their inverses r . The same is i i ˆ ˆ ¯ 9 9 true for GPLH’s. For example, if GPL l 9 is given by g 5 p ∨ p p then GPLH L 9 is ˆ l 9 1 1 3 ] ] 21 21 21 21 21 ˜ ˜ ˆ ˜ ˜ ¯ ¯ r ∨ r r . It can be transformed to L 9 5 r ∨ r r r 5 r ∨ r r , where r 5 r r 1 1 3 1 1 1 3 1 1 3 1 1 1 ˆ is the incomparability relation for r . The social relation r 5 L 9r , r , r is described 1 1 2 3 ˜˜ by xry ⇔ xr y ∨ xr y ∧ yr x. 1 1 3 For operator Fr , . . . , r we will denote by J the set of all indices j of its essential 1 n F variables r a variable r is unessential if any variation of r preserves the value of the i i i J operator. A PLH L9 with J 5 J is referred to as LH on J and is denoted by L . A L9 J ˆ 9 9 GLH L on J is defined analogously. If g 5 w p , . . . , p , p , . . . , p is the function F 1 n 1 n corresponding to F then the dual operator F can be introduced by g 5 g . F F Applying the theorem from Section 7 to the results of Section 6, we can obtain the n explicit form of operators 0 → 5 for all classes 5 0 from 3 Sholomov, 1990b; Sholomov, 1994. Statements b, c of Section 6 lead to the proposition 1 ˆ a F [ F 0, 5 , 5 [ h0, 6, j, iff F is equivalent to a GPLH L9. If F [ F 0, 5 then F is equivalent to some PLH L9. For the class 5 5 0 and monotone operators, the result was published by Fishburn 1975. The article Wilson, 1972 allows us to predict the fact for the case of 5 5 0 and of general type operators strict proofs for the case are in Aleskerov and Vladimirov 1986, Levchenkov 1990 and Morkjalunas 1985. For 5 [ h6, j and monotone 98 L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 operators the result could be predicted on the basis of Blair and Pollak 1979 and Blau 1979, the explicit formulation for 5 5 6 and monotone operators is given by Danilov 1983. Complete description of all mappings into the class of partial orders results from d of Section 5. ˆ b F [ F 0, 3 iff F is equivalent to an intersection of GLH ’s L of PLH’s 1i k i 1 9 L if F [ F 0, 3 . 1i k i This result for monotone operators is obtained by Danilov 1983. In nonmonotone case the synthesis problem was solved by Vladimirov 1987 and Levchenkov1990, but our solution gives a more restricted set containing for each operator of the type n → 3 an equivalent operator. Note that it is impossible to strengthen this result for monotone operators by using LH’s instead of PLH’s. That can be done iff F satisfies the Pareto condition Danilov, 1983; also see Remark in Section 6. Some analogy between n the form of operators 0 → 3 and Dushnik–Miller theorem see Ore, 1962 was noted in Danilov 1983 for the monotone case. A general form of aggregation operator for the class of transitive relations follows from d and e of Section 6. ˆ c F [ F 0, 7 iff F is equivalent to an operator in either of two forms L 1i k i J J J 1 ˆ ˆ 9 or L of L or L if F [ F 0, 7 where all L are 1i k i 1i k i 1i k i i GLH ’s on the same J. The synthesis problem for F 0, 7 was solved by Vladimirov 1987 and Levchenkov 1990, but we indicate a smaller set containing for each F [ F 0, 7 an equivalent operator. The statement g of Section 6 makes it possible to indicate all operators which map collections of weak orders to acyclic relations. ˆ d F [ F 0, iff F is equivalent to an operator expressible in the form L G in 1 1 1 1 the form L G if F [ F 0, where G [ F G [ F is arbitrary. In an equivalent form, this result is in Vladimirov 1987. For monotone operators, the result is contained in Blau and Deb 1977. In the case of nonmonotone operators, some weaker assertion is obtained in Kelsey 1984. Denote by ‡ and , respectively, the classes of asymmetric and complete relations. From results of Section 6 it follows that operators of F 0, ‡ and F 0, are equivalent to operators, expressible in the forms F 5 G G and F 5 G G , respectively, where G is any operator any monotone operator, in the monotone case. In t addition, in Sholomov 1990b a general presentation for operators of F 0, is is t found, where is the class of t-acyclic relations. The approach, based on the theorem from Section 6, allows us to simplify proofs of n some known results concerning neutral operators 0 → 5. As an illustration, let us n derive the characterization of operators 0 → which was obtained by Ferejohn and Fishburn 1979 and somewhat simplified in Aleskerov and Vladimirov 1986. In our terms, the result has the following formulation: Fr , . . . , r [ ⁄ F0, iff there exist 1 n n ˜ ˜ k and s , . . . , s [ h 2 1, 0, 1 1j for which 1 k ˜ ˜ g s 5 ? ? ? 5 g s 5 1, 24 F 1 F k 1 2 ˜ ˜ v s 5 v s , 25 1i k i 1i k i 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 +