Rings and Fields, a Constructive View

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.

Earlier papers [BV22, BV23] introduced the notions of a core and an index of a relation (an index being a special case of a core). A limited form of the axiom of choice was postulated-specifically that all partial equivalence relations (pers) have an index-and the consequences of adding the axiom to axiom systems for point-free reasoning were explored. In this paper, we define a partial ordering on relations, which we call the thins ordering. We show that our axiom of choice is equivalent to the property that core relations are the minimal elements of the thins ordering. We also postulate a novel axiom that guarantees that, when thins is restricted to non-empty pers, equivalence relations are maximal. This and other properties of thins provide further evidence that our axiom of choice is a desirable means of strengthening point-free reasoning on relations. Although our novel axiom is valid for concrete relations and is a sufficient condition for characterising maximality, we show that it is not a necessary condition in the abstract point-free algebra. This leaves open the problem of deriving a necessary and sufficient condition.

Logic Journal of IGPL, 2007

We discuss the question of inclusions between positive relational terms and some of its aspects, using the form of a dialogue. Two possible approaches to the problem are emphasized: natural deduction and graph manipulations. Both provide sound and complete calculi for proving the valid inclusions, supporting nice strategies to obtain proofs in normal form, but the latter appears to present several advantages, which are discussed. 1 2 On Positive Relational Calculi system. We formulate the rules and prove they are sound and complete. We also present a strategy for finding proofs using the natural deduction rules. Then, in Section 4, we introduce a third approach: a graph calculus. This is a refinement of the proof strategy of the natural deduction system into a simple and playful formal system. In presenting the graph calculus, we concentrate mainly on its intuitive aspects. Historical and technical details are presented elsewhere , where a list of references can be found. Finally, Section 5 discusses some ongoing work and perspectives.

The Review of Modern Logic, 2004

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.

Algebra Universalis, 1995

We solve a problem of J6nsson by showing that the class Y/of (isomorphs of) algebras of binary relations, under the operations of relative product, conversion, and intersection, and with the identity element as a distinguished constant, is not axiomatizable by a set of equations. We also show that the set of equations valid in ~ is decidable, and in fact the set of equations true in the class of all positive algebras of relations is decidable.

Publications de l'Institut Math?matique (Belgrade), 2013

Following Jiang Guanghao and Xu Luoshan?s concept of conjugative, dually conjugative, normal and dually normal relations on sets, the concept of quasi-regular relations is introduced. Characterizations of quasi-regular relations are obtained and it is shown when an anti-order relation is quasiregular. Some nontrivial examples of quasi-regular relations are given. At the end we introduce dually quasi-regular relations and give a connection between these two types of relations.