Explicit form of operators 7

102 L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 t 5 w 1 ? ? ? 1 w 2. After the normalization of w , . . . , w , t, the realization is 1 n 1 n reduced to w , . . . , w , 1 2 where w 1 . . . 1 w 5 1. 1 n 1 n To each monotone Boolean function f 5 k x . . . x , i i 1 s hi , . . . ,i j 1 s put in correspondence the function 9 u 5 k n p n p f i j hi , . . . , i j i [hi , . . . , i j j [ ⁄ hi , . . . , i j 1 s 1 s 1 s of P , and denote by Q the aggregation operator associated with u . Also denote by 3,2 f f o 7 the set of operators Q for all f [ Th , and by 7 the set of operators similar 1 2 f 1 2 1 2 o to operators of 7 a similar operator comes out the initial one by replacements of 1 2 21 o some r by r . The sense of operators Q [ 7 can be explained in the following i i f 1 2 manner. Each individual i, 1 i n, has w votes, and the alternative x is preferred over i y in the collective decision iff none of voters prefers y and at least one half of votes o support x. According to that, operators of the class 7 of 7 will be called 1 2 1 2 generalized weighted majority operators with right of veto. The following statement is valid. d F [ F 6, iff F is equivalent to an operator Q G where Q [ 7 , G [ F 1 2 o 1 1 Q [ 7 , G [ F , if F [ F 6, . 1 2 n

11. Explicit form of operators 7

→ 7 Our description of the class F 7, 7 Sholomov, 1998b uses the following special operators which have not appeared in the scientific literature before: T r , . . . , r 5 r r , n 1 29 S D n 1 n i j 1i n 1j i 1 T r , . . . , r 5 r r r , n 1. 30 S D n 1 n i j j 1i n 21 1j i 1j n Their logic presentations 9 are T r , . . . , r 5 r r ∨ r r r ∨ ? ? ? ∨ r , . . . , r r r n 1 n 1 1 1 2 2 1 n 21 n n 1 T r , . . . , r 5 r r ∨ r r r ∨ ? ? ? ∨ r , . . . , r r r ∨ r , . . . , r . n 1 n 1 1 1 2 2 1 n 22 n 21 n 21 1 n 1 It can be proved directly that the operators T 5 r r and T 5 r r r 1 1 1 2 1 1 1 r preserve transitivity i.e. transform TR’s to TR’s. Since all operators 29, 30 are 2 1 expressible as some superpositions of T and T , the fact can be extended to arbitrary 1 2 1 T and T . n n 1 The operators T and T are ‘‘transitive analogs’’ of LH L 13 and of L , dual to n n n n a 21 LH. To compare the operators, let us associate to a relation r the relations r 5 r\r 5 L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 103 ] s 21 21 ˜˜ ¯ r r asymmetric, r 5 r r symmetric, r 5 r r of incomparability. By means of equivalent transformations, the operators L and T can be represented as n n ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ L 5 r ∨ r r ∨ ? ? ? ∨ r r r r , n 1 1 2 1 2 n 21 n a s a s s s a T 5 r ∨ r r ∨ ? ? ? ∨ r r . . . r r n 1 1 2 1 2 n 21 n 1 the accordance between L and T is similar. The operators 29, 30 will be called n n transitive LH’s TLH’s. Note, in contrast to TLH, LH do not preserve transitivity. An 21 operator, obtained from a TLH by substituting r instead of r for some indices i, will i i be called a generalized TLH GTLH. Similarly to PLH and GPLH see Section 8, we will define a partial TLH PTLH and a generalized PTLH GPTH. As the relation 21 n inverse operation preserve transitivity, all defined operators have type 7 → 7. The following statement Sholomov, 1998b describes explicitly the operators preserving transitivity. ˆ ˆ ˆ a F [ F 7, 7 iff F is equivalent to an operator T , where T , . . . ,T are 1i k i 1 k 1 GPTLH ’s PTLH’s, if F [ F 7, 7 . The operators of a are in the class F 7, 7 because the operation preserves transitivity. The proof of the inverse proposition is considerably harder. Let us give its outline. n For all preceding types of operators 5 → 5 , the class 5 contained asymmetric 1 2 1 relations only. It allowed us to describe operators F in terms of functions g [ P 22. F 3,2 If relations of 5 are not asymmetric in particular, at 5 5 7 it is necessary to use the 1 1 more general logic form 9. Let the function w in 9 be given by a formula in the basis h ∨ , ∧ , ∨ ¯ j with negations applied to elementary variables only. Eliminating the negations in 9 by means of the ] 21 21 ¯ equivalent replacements r 5 r and r 5 r , one can transform the formula 9 i i i i to the form 21 21 21 21 Fr , . . . , r 5 cr , . . . , r , r , . . . , r , r , . . . , r , r , . . . , r , 31 1 n 1 n 1 n 1 n 1 n 21 21 where c is a formula in the basis h ∨ , ∧ j. Call to mind, the symbols r , r , r , r i i i i are considered as Boolean variables which take values 1 or 0 for each concrete x, y. The approach, worked out in Section 5 for the study of transitive ordinal relations and n based on a special operation of composition, can be adapted to operators 7 → 7. Let us introduce the suitable notions. The conjunction K 5 q . . . q is called an elementary conjunction if each factor q , 1 n i 21 21 21 1 i n, is of one of following type: a r , b r , c r r , d r , e r , f i i i i i i 21 21 21 r r , g r r , h r r , j 1 q 5 1 means the absence of the suitable factor. i i i i i i i The factors of types a–j are called elementary factors . A disjunction K ∨ ? ? ? ∨ K of elementary conjunctions is called a disjunctive normal 1 s form DNF. Any function c in the form 21 can be presented by means of a DNF. Let us define a binary operation F + G of operators composition. As in Section 5, the operation will be described for elementary factors at first, for elementary conjunction after that, and for operators 31 in the form of DNF at last. 1 Let us introduce the following classes of elementary factors. The class Q consists of 104 L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107 2 all factors having types a or g, the class Q is formed by all factors of types b or ˆ h, and all factors of types d 2 f , j produce the class Q. The classes do not contain 21 9 the factors r r of type c only. The result of operation q + q on elementary factors q i i i i i ˆ ˆ 9 9 9 and q is determined in the following way. If q [ Q or q [ Q then q + q 5 1. Now, let i i i i i 1 2 ˆ 9 9 q , q [ ⁄ Q. In this case, if q and q are contained in the same class Q or Q then i i i i 1 2 9 9 9 q + q 5 q ∧ q . If they belong to different classes Q and Q then q + q 5 1. If some i i i i i i 21 ˆ 9 9 9 of the factors q ,q coincides with r r and q , q [ ⁄ Q then the result of q + q is i i i i i i i i equal to the other of the factors. 9 9 For conjunctions K 5 q , . . . , q , K9 5 q , . . . , q and DNF’s c 5 K ∨ ? ? ? ∨ K , 1 n 1 n 1 s 9 9 c9 5 K ∨ ? ? ? ∨ K , the result of operation K + K9 and c + c9 are defined by 14 and 1 t 15. For the operators F 5 c and G 5 c9, we set F + G 5 c + c9 it can be proved that the result of composition operation F + G does not depend on chosen DNF’s c and c9. Any operator F satisfying the condition F + F 5 F is referred to as closed with respect to composition. The role of the composition operation is clarified by the following assertion: F [ F 7, 7 iff F is closed . In this way, the explicit description of all operators n 7 → 7, mentioned in the statement a, can be found.

12. Summary table