Techniques for study of ordinal relations

L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 89 ˜ ˜ ˜ as the disjunction of K u for all s such that gs 5 1 if g ; 0 put w ; 0. It is easy to ˜ s prove for monotone g that w can be taken to be monotone. Denoting pu 5 p , i i ˜ ˜ 9 9 9 p9u 5 p , p 5 p , . . . , p , p 9 5 p , . . . , p , we will write 11 as i i 1 n 1 n ˜ ˜ ˜ gu 5 wp, p 9. 12 As examples, let us present the describing functions of both the lexicography and the Pareto relation: 9 9 9 9 g 5 p ∨ p p ∨ ? ? ? ∨ p p ? ? ? p p , 13 l 1 1 2 1 2 n 21 n 9 9 9 g 5 p p ? ? ? p p ∨ p ∨ ? ? ? ∨ p p 1 2 n 1 2 n the symbols ∧ of conjunction are omitted. ¯ ¯ ¯ 9 9 9 A conjunction K 5 q . . . q is called elementary if q [ h p , p , p , p , p p , 1j, 1 n i i i i i i i 1 i n q 5 1 indicates the absence of ith factor. A disjunction of elementary i conjunctions is called a disjunctive normal form DNF. The formula 13 is an example of DNF. Any function g [ P can be presented by some DNF for instance, as described 3,2 above.

5. Techniques for study of ordinal relations

It is possible to use, instead of operations with ordinal relations, operations with their describing functions Sholomov, 1990a; Sholomov, 1994. 5.1. Set-theoretical operations on ordinal relations If r 5 Fr , . . . , r , r, r , . . . , r are ordinal relations, F is a s.-t. operation, then 1 k 1 k ˜ ˜ ˜ g u 5 w g u , . . . , g u where the Boolean function w corresponds to F. This r r r 1 k fact is evident. ¯ As an example, let us consider the relation l\p 5 l p the complement of the 2 Pareto relation with respect to the lexicography on R . We have ]]]] ] ] ¯ ¯ ¯ ¯ 9 9 9 9 9 9 9 g 5 g g 5 p ∨ p p p p p ∨ p 5 p ∨ p p p ∨ p ∨ p p 5 p p . l\p l p 1 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 Thus, x l\py ⇔ x . y ∧ x , y . 1 1 2 2 5.2. Inversion of ordinal relation 21 ˜ The inverse r of an ordinal relation r is described by the function g u 5 21 r ˜ ˜ g 2u , 2 u 5 2u , . . . , 2 u . It arises from r 1 n 21 g Dx,y 5 1 ⇔ x r y ⇔ y rx ⇔ g Dy,x 5 1 ⇔ g 2 Dx,y 5 1. 21 r r r ˆ The relation r, formed from the ordinal relation r by the inversion of directions for n 21 some axes in R , is said to be similar to r the inversion of all axes gives r . Because 90 L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 ¯ ¯ of p2u 5 p 9u and p92u 5 pu, a presentation of g can be obtained from the ˆ r ¯ ¯ 9 9 presentation 12 of g by substitutions of p and p instead of p and p , respectively, r i i i i for variables corresponding to inverted axes. 2 For example, describing function of the Pareto relation inverse on R is g 5 21 p ˆ ¯ ¯ ¯ ¯ 9 9 p p p ∨ p , while the relation l formed from the lexicography l by inversion of the 1 2 1 2 ¯ 9 9 second axis is described by the function g 5 p ∨ p p . ˆ l 1 1 2 5.3. Dual ordinal relation ¯ ¯ ¯ By analogy to the notion of dual Boolean function w x , . . . , x 5 wx , . . . , x , 1 n 1 n ˜ ˜ ˜ ˜ ¯ we define the dual function g u for gu [ P setting g u 5 g2u . From 3,2 ] 21 Sections 5.1, 5.2 and r 5r it follows 21 ˜ ˜ ˜ ˜ ¯ ¯ g u 5 g u 5 g 2u 5 g u . r r r r ˜ So the describing function of a dual relation r is dual to g . If gu is presented in the r ˜ ˜ ˜ ˜ ˜ form 12 then g u 5 w p 9, p p and p 9change places. It arises from ˜ ˜ ¯ ¯ 9 9 g u 5 g2u 5 w p 2u, . . . , p 2u, p 2u, . . . , p 2u 1 n 1 n ˜ ˜ ¯ ¯ ¯ ¯ ¯ 9 9 5 wp u, . . . , p u, p u, . . . , p u 5 w p 9, p . 1 n 1 n Recall, if the function w is given by a formula in the basis h ∨ , ∧ , ∨ ¯ j then a formula for w can be formed by replacements of all ∨ and ∧ by ∧ and ∨ , respectively. 2 For example, the describing functions of relations on R dual to p and l are 9 9 9 9 g 5 p p p ∨ p 5 p ∨ p ∨ p p , p 1 2 1 2 1 2 1 2 9 9 9 9 9 g 5 p ∨ p p 5 p p ∨ p 5 p ∨ p p . l 1 1 2 1 1 2 1 1 2 5.4. Product of ordinal relations A relation r is the product of relations r and r r 5 r r if xry ⇔ zx r z ∧ z r y 1 2 1 2 1 2 holds for all x, y. The product operation on ordinal relations can be described in terms of ˆ ˆ a binary operation of composition g + g on functions g, g [ P . We will describe the 3,2 ˆ ¯ ¯ ¯ 9 9 9 composition operation sequentially: for factors q , q [ h p , p , p , p , p p , 1j at first, i i i i i i i i ˆ ˆ for conjunctions K, K after that, and for functions g, g in the form of DNF at last. For ˆ factors, the operation q + q is defined by Table 1, rows and columns of which are i i Table 1 ¯ ¯ ¯ p p9 p p 9 p9p 1 p p p 1 1 p 1 p9 p p9 1 1 p9 1 ¯ ¯ ¯ ¯ p 1 1 p p 9 p 1 ¯ ¯ ¯ ¯ p 9 1 1 p 9 p 9 p 9 1 ¯ ¯ ¯ ¯ p9p p p9 p p 9 p9p 1 1 1 1 1 1 1 1 L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 91 ˆ associated with factors q and q , respectively subscripts i in the table are omitted. For i i ˆ ˆ ˆ elementary conjunctions K 5 q , . . . , q and K 5 q , . . . , q we put 1 n 1 n ˆ ˆ ˆ K + K 5 q + q ∧ ? ? ? ∧ q + q 14 1 1 n n ˆ ˆ ¯ ¯ ¯ ¯ ¯ 9 9 9 9 9 for example, if K 5 p p p p , K 5 p p p p , then K + K 5 p + p ∧ p p + p ∧ 1 2 2 3 1 2 3 4 1 1 2 2 2 ¯ ˆ 9 p + p ∧ 1 + p 5 1 ∧ p ∧ p ∧ 1 5 p p . For functions g 5 K ∨ ? ? ? ∨ K and g 5 3 3 4 2 3 2 3 1 u ˆ ˆ K ∨ ? ? ? ∨ K , presented in the form of DNF, we define the operation as 1 v ˆ ˆ g + g 5 k K + K . 15 s t 1s u, 1t v ˆ Formally, the result of the operation g + g depends on the used DNF’s. But it can be ˆ shown that different DNF of functions g and g lead to different DNF of the same ˆ function g + g, i.e. the dependence is fictitious. The following assertion explains the role of the operation. The describing function g of the product r r can be found as r r 1 2 1 2 g + g . r r 1 2 2 ˆ For example, let us find the product of the Pareto relation on R and the relation l 9 9 from 5.2. Taking into account that DNF of g is p p ∨ p p , we have p 1 2 1 2 ¯ ¯ 9 9 9 9 9 9 9 9 g 5 g + g 5 p p ∨ p p + p ∨ p p 5 p p + p ∨ p p + p p ˆ ˆ p l p l 1 2 1 2 1 1 2 1 2 1 1 2 1 2 ¯ 9 9 9 9 9 9 ∨ p p + p ∨ p p + p p 5 p ∨ p ∨ p ∨ p 5 p . 1 2 1 1 2 1 2 1 1 1 1 1 Thus, xg y ⇔ x y . ˆ p l 1 1 ¯ ¯ ¯ Now, let us give a sketch of the proof. Associate with the factors p, p9, p, p 9, p9p, 1 the symbols . , , , , , 5 , [, respectively. Create Table 2 by replacement in Table 1 of each q [ h p, p9, . . . , 1j by the corresponding n [ h . , , . . . , [j. It easy to verify directly the following fact. If for some quantities x, y, z we take in Table 2 some ˆ ˆ ˆ ˚ row n and column n such that xnz and zny, then the cell isolated by n and n contains n ˚ such that x ny, where [ signifies that correlation between quantities can be arbitrary. In addition, any correlation between x and y, presented in the cell, is attainable for suitably chosen z. Let r and r be ordinal relations, x and y be such that g Dx,y 5 1 i.e. 1 2 r r 1 2 x r r y. Then xr z and zr y hold for some z, from where g Dx,z 5 1 and 1 2 1 2 r 1 ˆ ˆ ˆ g Dz,y 5 1. There are conjunctions K 5 q , . . . , q and K 5 q , . . . , q of DNF’s g r 1 n 1 n r 2 1 ˆ and g such that K Dx,z 5 1 and KDz,y 5 1. It is not difficult to understand that r 2 ˆ ˆ ˆ x n z and z n y take place with n and n corresponded to q and q , 1 i n. Hence, the i i i i i i i i i i ˆ ˚ ˚ correlations x n y , 1 i n, hold where n is corresponded to q + q . It leads to i i i i i i Table 2 . , 5 [ . . . [ [ . [ . [ [ [ [ [ , [ , [ [ , , , [ 5 . , 5 [ [ [ [ [ [ [ [ 92 L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 ˆ ˜ K + K Dx,y 5 1 and gives g + g Dx,y 5 1. As a result, we obtain g u 5 r r r r 1 2 1 2 ˜ ˜ ˜ 1 ⇒ g + g u 5 1. The inverse proposition g + g u 5 1 ⇒ g u 5 1 can be r r r r r r 1 2 1 2 1 2 derived in a somewhat similar way. The composition operation has a number of good properties. It is symmetric, associative, monotone g9 g ⇒ g9 + g0 g + g0. 5.5. Transitive closure of ordinal relations Because of associativity of the composition operation, the many-placed operation s g + ? ? ? + g can be introduced. Let g be a s-tuple composition g + g + ? ? ? + g. It is not 1 s 2 s ˜ difficult to prove that g g ? ? ? g . . . the notation g9 g0 means g9u n s ˜ ˜ g0u for all u [ h 2 1,0, 1 1j . Since the sequence hg j is increasing and bounded s 11 s s 9 above, it will be g 5 g beginning with some s9. Denote by [g] the result g of the stabilization. The following fact is fairly evident. If [ r] is the transitive closure of an 3 ordinal relation r then g 5 [g ]. As an example, consider on R the relation r given [ r ] r 2 3 9 9 by the function g 5 p p ∨ p p . One can verify directly that g 5 p ∨ p p 5 g . r 1 2 1 3 r 1 1 3 r 9 Hence, g 5 p ∨ p p i.e. [ r] is the lexicography on 1st and 3rd components. [ r ] 1 1 3 ˜ ˜ ˜ ˜ ¯ ¯ 9 9 Put 0 5 0, . . . , 0 and D 5 p ∨ p ∨ ? ? ? ∨ p ∨ p note, D u 5 1 ⇔ u ± 0 . The 1 1 n n next assertion characterizes properties of ordinal relations r in terms of functions g . r An ordinal relation r is a reflexive, b irreflexive, c asymmetric, d antisymmet- ˜ ric , e complete, f connected, g transitive, h negative transitive iff a g 0 5 1, r ˜ b g 0 5 0, c g g , d D g D g , e g g , f D g D g , g r r r r r r r r r g + g g , h g + g g . r r r r r r ˜ Item a follows from x rx ⇔ g Dx,x 5 1 ⇔ g 0 5 1; b is similar to a. Note, a r r and b imply each ordinal relation to be reflexive or irreflexive. Let us prove c. If r is asymmetric then ¯ g Dx,y 5 1 ⇒ g Dy,x 5 0 ⇒ g 2 Dx,y 5 1 ⇒ g Dx,y 5 1. r r r r As x, y are arbitrary, g g holds. Conversely, g g implies r r r r ¯ g Dx,y 5 1 ⇒ g Dx,y 5 1 ⇒ g 2 Dx,y 5 1 ⇒ g Dy,x 5 0. r r r r It signifies that r is asymmetric. Item e is dual to c. As a distinction between ˜ ˜ asymmetry and antisymmetry appears at u 5 0 only, the statements c and e imply d 2 and f. Taking into account the equivalence of transitivity of r to r r, we obtain g. Item h is dual to g. The described techniques allow us to solve a number of principal problems in the design and the analysis of multicriteria choice models Sholomov, 1990a; Sholomov, 1990b; Sholomov, 1994; Sholomov, 1996. Earlier, similar problems were considered in Berezovsky et al. 1989 and Makarov et al. 1987 where some qualitative results were obtained but no efficient methods were worked out. Also, tools of this section make it possible to find the explicit description of ordinal relations for all classes in 3. L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 93

6. Characterization of ordinal relations