Connection between aggregation operators and ordinal relations

L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 95 side can be supplemented to TL. Since the consideration was led up to reindexing and inverting of axes, we have g g , in the general case. ˆ r l The assertion a is a simple corollary of f. According to f, a LO r can be extended ˆ to some GL l. As a connected relation r cannot be increased without loss of asymmetry, ˆ r coincides with l. Conversely, any GL meets the conditions b, c, f of Section 5, and therefore arbitrary GL is a LO. n The results of this section show that the structure of ordinal relations over R is fairly simple. Namely, all LO’s are reduced to lexicographies up to directions of coordinate n axes; WO’s are reduced to lexicographies on subspaces of R ; proper SO’s and proper IO’s are missing; each PO is an intersection of lexicographies; TR’s are equivalent to n PO’s strict or not on subspaces of R ; each AR is a subset of some lexicography.

7. Connection between aggregation operators and ordinal relations

All classes of relations in 3 except for 7 consist of asymmetric relations only. We will suppose up to Section 11 devoted to 7 individual relations to be asymmetric. The 21 condition of asymmetry r r 5 5 is equivalent to the inclusion r r , i.e. to the i i i i inequality xr y xr y. i i Let Fr , . . . , r 5 Fr , . . . , r , r , . . . , r be an operator of F. Associate with F 1 n 1 n 1 n ˆ ˆ the function g [ P as below. To find the value g for some u , . . . , u , consider F 3,2 F 1 n any x, y [ A and any profile of asymmetric relations r , . . . , r such that 1 n ˆ yr x if u 5 2 1, i i ¯ ¯ ˆ xr y ∧ yr x if u 5 0, , 19 i i i 6 ˆ xr y if u 5 1, i i and put 1 if x, y [ Fr , . . . , r , 1 n ˆ ˆ g u , . . . , u 5 20 H F 1 n if x, y [ ⁄ Fr , . . . , r . 1 n It is easy to see that the definition is correct i.e. independent of chosen x, y and r , . . . , 1 r provided 19. The function g was used in Aleskerov and Vladimirov 1986. n F It is not difficult to understand that the map F → P is injective F ± F 9 ⇒ g ± g . 3,2 F F 9 Let us prove that it is also surjective and hence bijective. Consider any g [ P , 3,2 9 9 gu , . . . , u 5 w p , . . . , p , p , . . . , p . 21 1 n 1 n 1 n Let F be the operator 8 with the function w from 21. Let us verify that g 5 g. The F ˆ ˆ equalities 20 and 21, with regard to the correlations pu 5 xr y and p9u 5 xr y, i i i i equivalent to 19, give ˆ ˆ g u , . . . , u 5 1 ⇔ x, y [ Fr , . . . , r ⇔ wxr y, . . . , xr y, xr y, . . . , xr y 5 1 F 1 n 1 n 1 n 1 n ˆ ˆ ˆ ˆ ˆ ˆ ⇔ w pu , . . . , pu , p9u , . . . , p9u 5 1 ⇔ gu , . . . , u 5 1. 1 n 1 n 1 n The correspondence between F and g in the explicit form is F 96 L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 9 9 F 5 Fr , . . . , r , r , . . . , r ⇔ g 5 w p , . . . , p , p , . . . , p . 22 1 n 1 n F 1 n 1 n Different presentations 21 of the same function g led to different ones of the same F operator F. Therefore, equivalent transformations in the class F are similar to them in P . 3,2 Associate with each operator F the ordinal relation r by g 5 g the corre- F r F F spondence is one-to-one. The following statement Sholomov, 1990b; Sholomov, 1994 n helps to find the explicit form of operators 0 → 5 for different 5. A particular case n of the approach for operators 0 → 0 was used in Fishburn 1975 and in Morkjalunas, 1985. Theorem. If 5 is an universally axiomatizable class then an operator F [ F has the n type 0 → 5 iff r [ 5. F A proof of the fact uses the following lemma where by A we mean a finite set. Lemma. If F is an operator in the form 8 then n a ;x . . . ;x [ R Ar . . . r [ 0Ax . . . x [ A 1 s 1 n 1 s x rx ⇔ x r x , 1 i, j s, r 5 Fr , . . . , r , i j i F j 1 n n b ;A;r . . . ;r [ 0A;x . . . ;x [ Ax . . . x [ R 1 n 1 s 1 s x rx ⇔ x r x , 1 i, j s, r 5 Fr , . . . ,r . i j i F j 1 n i i Proof of Lemma. a Let x 5 x , . . . , x , 1 i s. Associate x with an element x i 1 n i i e.g. an integer and put A 5 hx , . . . , x j. Introduce the relations r 1 t n on A by 1 s t i j ˆ setting x r x ⇔ x . x , 1 i, j s. Consider any i, j, and denote Dx , x by u , . . . , i t j t t i j 1 ˆ ˆ ˆ ˆ ˆ 9 u . Introducing p 5 pu and p 5 p9u , we have n t t t t i j ˆ x r x ⇔ x . x ⇔ p 5 1, i t j t t t i j ¯ ˆ 9 x r x ⇔ x r x ⇔ x x ⇔ p 5 1. i t j j t i t t t With regard to 8 and 22, this gives ˆ ˆ ˆ ˆ 9 9 x rx ⇔ wx r x , . . . , x r x , x r x , . . . , x r x 5 1 ⇔ wp , . . . , p , p , . . . , p 5 1 i j i 1 j i n j i 1 j i n j 1 n 1 n ˆ ˆ ⇔ g u , . . . , u 5 1 ⇔ g Dx , x 5 1 ⇔ x r x . F 1 n F i j i F j b Since r [ 0A 1 t n, the finite set A can be subdivided into levels such that t i xr y holds iff x’s level is higher than y’s level Fishburn, 1970. Denote by x the t t i i 9 number of the x s level in r , and put x 5 x , . . . , x , 1 i s. We have i t i 1 n i j x r x ⇔ x . x , i.e. x and x connect with x and x by the same way as in a. It i t j t t i j i j makes it possible to finish the proof similarly to a, and to obtain x rx ⇔ x r x , 1 i, i j i F j j s. 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