L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107
83
detailed references will be given and has brought some new outcomes. Especially, we
n n
mention the cases of mapping 6 →
and 7 →
7 which led to some kinds of operators not encountered in the scientific literature before. This approach also helps to
construct the SDR for a number of specific choice problems for example, to take into account a priority of individuals; see Section 13.
The revealed correspondence between aggregation operators and ordinal relations allows us to find connections between results on aggregation and known facts of the
theory of relations and to explain the observed analogies. The proposed approach simplifies the study of SDR by reduction of fairly complicated objects operators on a set
of relations to the simpler objects ordinal relations. This allows us to use the geometric intuition in SDR search.
By the way, we worked out effective tools for operating with ordinal relations. The means are of great interest for the theory of multicriteria choice which widely uses
ordinal relations Berezovsky et al., 1989; Makarov et al., 1987. All proofs are based on the special developed techniques for two-valued functions in
three-valued variables. Other logic methods were used in Aleskerov and Vladimirov, 1986; Levchenkov, 1990; Murakami, 1968; Vladimirov, 1987 for aggregation problems,
in Berezovsky et al., 1989; Makarov et al., 1987 for ordinal relations analysis, and in Sholomov, 1989 for different problems relating to discrete models of choice.
2. Classes of relations
2
Let r be a binary relation on a given set A r A . For x, y [ A, the notations xry ¯
and xry mean x, y [ r and x, y [ ⁄ r, respectively. The relation r is referred to as
19 reflexive if xrx holds,
¯ 29
irreflexive if xrx holds, ¯
39 asymmetric if xry
⇒ yrx,
¯ 49
antisymmetric if x ± y ∧
xry ⇒
yrx, 59
complete if xry ∨
yrx holds, 69
connected if x ± y ⇒
xry ∨
yrx, 79
transitive if xry ∧
yrz ⇒
xrz, ¯
¯ ¯
89 negative transitive if xry
∧ yrz
⇒ xrz.
]
21 2
21 21
The relation r 5r 5 A \r
is called dual to r r is the relation inverse to r:
21
xr y
⇔ yrx. It is easy to see that pairs of conditions 19–29, 39–59, 49–69,
84 L
.A. Sholomov Mathematical Social Sciences 39 2000 81 –107
79–89 are pairwise dual i.e., for example, if r satisfies the condition of asymmetry then r
is a complete relation and vice versa. A transitive, irreflexive relation is called a strict partial order PO; by a linear
order LO we mean a connected PO; a negative transitive, asymmetric relation is called a weak order WO. A PO r provided that
xry ∧
zrv ⇒
xrv ∨
zry 1
is referred to as an interval order IO; an IO r restricted by the condition xry
∧ yrz
⇒ xrv
∨ v
rz 2
is called a semiorder SO . A relation r which has no cycles x rx
∧ x rx
∧ ? ? ?
∧
1 2
2 3
x rx
∧ x rx , s 1, is said to be acyclic
.
s 21 s
s 1
We denote by 3, +, 0, , 6, and 7 the sets of all PO’s, LO’s, WO’s, IO’s, SO’s, acyclic relations AR’s and transitive relations TR’s, respectively, on arbitrary sets A.
In the case of relations on the given set A we will use the notations 3A, +A, . . . . The following strict inclusions
+ , 0 , 6 , , 3 , 7 3
are valid the classes 7 and are incomparable and both of them contain 3. All principal types of relations used in the choice theory are present in 3.
Usually, classes of relations are described by means of axioms. One can consider any binary relation as the binary predicate rx, y 5 xry on A. In the majority of the cases,
axioms which describe meaningful properties of relations can be written in the form ;x . . . ;x Px , . . . , x ,
4
1 s
1 s
for some integer s. Here the formula P includes, besides logic operations, the single predicate symbol r. According to terminology of Malcev 1973, we will call a class of
relations universally axiomatizable if it can be described by means of axioms 4. A set of axioms is not supposed to be finite. For example, the class of acyclic relations is
defined by the following countable set of axioms
] ;x . . . ;x x rx nx rx n . . . nx
rx ⇒
x rx , y 5 1, 2, . . .
1 y
1 2
2 3
y 21 y
y 1
It is essential that P does not contain equality signs, i.e. 4 is a formula of the Pure predicate calculus Malcev, 1973. Let us indicate a property of universally axiomatiz-
able classes which will be needed later on. It will clarify partly why equalities are inadmissible.
2
We say that a relation r9 is formed by a fission of the given relation r A at the
2
element a [ A if r9 A9 where A9 5 A
ha9j for some a9 [⁄ A, and
r9 5 r ha9,yuaryj hz,a9uzraj r
5
a 9
¯ where r 5
ha9,a9j if ara, and r 5 5 if ara. The fission operation is inverse to a
a 9 a 9
identification: the relation r can be formed from r9 by identifying a and a9. If a relation is obtained from r by finite number of successive fissions at some elements, we will
L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107
85
say that it is formed by fission of r any WO on a finite set can be formed by fission of some LO.
A class relations 5 will be called closed with respect to fission if for each r [ 5 it contains all relations formed by fission of r. Any universally axiomatizable class 5 is
closed with respect to fission . To make sure of that, consider a formula 4, used in the
definition of 5. For any given x , . . . , x [ A9, examine the value of P on the relation r9
1 s
5, formed by fission of r [ 5. According to 5, replacements in P of all r9 and a9 by r and a do not change its value. The new value is true as it relates to r [ 5. Hence, the
original value of P on r9 is also true. It gives r9 [ 5. All properties of relations introduced above except for 49 and 69 which use
inequalities negations of equalities can be described in terms of axioms 4, and all classes in 3 except for + are universally axiomatizable. Note, the class + is nonclosed
with respect to fission.
3. Aggregation operators
