Rings and Fields, a Constructive View

Annals of Operations Research, 2015

Outranking relations such as produced by the Electre I or II or the Tactic methods are based on a concordance and non-discordance principle that leads to declaring that an alternative is "superior" to another, if the coalition of attributes supporting this proposition is "sufficiently important" (concordance condition) and if there is no attribute that "strongly rejects" it (non-discordance condition). Such a way of comparing alternatives is rather natural and does not require a detailed analysis of tradeoffs between the various attributes. However, it is well known that it may produce binary relations that do not possess any remarkable property of transitivity or completeness. The axiomatic foundations of outranking relations have recently received attention. Within a conjoint measurement framework, characterizations of reflexive concordance-discordance relations have been obtained. These relations encompass those generated by the Electre I and II methods, which are non-strict (reflexive) relations. A different characterization has been provided for strict (asymmetric) preference relations such as produced by Tactic. In this paper we briefly review the various kinds of axiomatizations of outranking relations proposed so far in the literature. Then we analyze the relationships between reflexive and asymmetric outranking relations in a conjoint measurement framework, consolidating our previous work. Co-duality plays an essential rôle in our analysis. It allows us to understand the correspondence between the previous characterizations. Making a step further, we provide a common axiomatic characterization for both types of relations. Applying the co-duality operator to concordance-discordance relations also yields a new and interesting type of preference relation that we call concordance relation with bonus. The axiomatic characterization of such relations results directly from co-duality arguments.

Algebra Universalis, 1988

The purpose of this paper is to present certain results arising from a study of quasi-orderings (pre-orderings). We show that to each relation R _ X x Y there are associated unique largest quasi-orderings :r/(R) on X and err(R) on Y such that &(R)oR o err(R) = R; and we present formulas for these quasi-orderings. For a fixed pair of quasi-orders Jrt and :r2 we characterize the invertible relations (with respect to the units 1rl and ~r2) in terms of isomorphisms between *:rz and "3r2, where *:ri is the partial ordering naturally induced by :ri. In particular we show that the set of invertible relations with rc~ = ~r2 = rc is a group isomorphic to the group Aut (*~r) of automorphisms of *:r. We present these results in sections 1-3 in the framework of a general relation algebra.

Bull. Int. Math. Virtual Inst., 2019

This investigation is in the mathematics based on the Intuition-istic logic. A relation ρ is a coequality relation if it is consistent, symmetric and co-transitive. For a coequality relation ρ on a set X with apartness we analyze the family Cop(X) of all classes of the relation. Characteristics of this family allow us to introduce a new concept, 'co-partition' in set with apaerness-a specific family of proper subsets. In addition, a connection between the family of all coequality relations and the family of all co-partitions is given. At the end of this article, some examples and applications in the semigroups with apartness theory are given.

MAT-KOL, 2018

In this paper, the concepts of several new classes of relations on sets are presented introduced by this author in the previous five years. The following classes of relations have been introduced and partly described in several his articles: the class of quasi-regular, the class of quasi-conjugative, the class of quasi-normal and the class of normally conjugative relations.

This investigation is in the mathematics based on the Intuition-istic logic. A relation ρ is a coequality relation if it is consistent, symmetric and co-transitive. For a coequality relation ρ on a set X with apartness we analyze the family Cop(X) of all classes of the relation. Characteristics of this family allow us to introduce a new concept, 'copartition' in set with apaerness-a specific family of proper subsets. In addition, a connection between the family of all coequality relations and the family of all copartitions is given. At the end of this article, some examples and applications in the semigroups with apartness theory are given.

Working Document, 2022

We introduce the general notions of an index and a core of a relation. We postulate a limited form of the axiom of choice ---specifically that all partial equivalence relations have an index--- and explore the consequences of adding the axiom to standard axiom systems for point-free reasoning. Examples of the theorems we prove are that a core/index of a difunction is a bijection, and that the so-called ``all or nothing'' axiom used to facilitate pointwise reasoning is derivable from our axiom of choice. We reformulate and generalise a number of theorems originally due to Riguet on polar coverings of a relation. We study the properties of the ``diagonal'' of a relation (called the ``diff\'{e}rence'' by Riguet who introduced the concept in 1951). In particular, we formulate and prove a general theorem relating properties of the diagonal of a relation to block-ordered relations; the theorem generalises a property that Riguet called an ``analogie frappante'' between the ``diff\'{e}rence'' of a relation and ``relations de Ferrers'' (a special case of block-ordered relations).

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1988

In this paper we propose an axiomatic characterization of the set of linear orders using the concept of a choice rule, which assigns to each ordered pair of feasible alternatives and a reflexive binary relation, exactly one element from the feasible set.

Science of Computer Programming, 2015

Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction techniques, and of other formal tools for checking key properties such as confluence, termination, and sufficient completeness. Such specifications would instead be amenable to formal analysis if an equationally-defined equality predicate enriching the algebraic data types were to be added to them. Furthermore, having an equationally-defined equality predicate is very useful in its own right, particularly in inductive theorem proving. Is it possible to effectively define a theory transformation E → E that extends an algebraic specification E to a specification E where equationally-defined equality predicates have been added? This paper answers this question in the affirmative for a broad class of order-sorted conditional specifications E that are sort-decreasing, ground confluent, and operationally terminating modulo axioms B and have subsignature of constructors. The axioms B can consist of associativity, or commutativity, or associativity-commutativity axioms, so that the constructors are free modulo B. We prove that the transformation E → E preserves all the just-mentioned properties of E. The transformation has been automated in Maude using reflection and it is used in Maude formal tools.