Árpád Száz

Basic tools and mild continuities in relator spaces

Starting with this paper, we offer a simple, unified foundation to general topology and abstract analysis by using a straightforward generalization of uniform spaces which leads us to more general structures than the ordinary topologies. As the extensive references show uniform spaces have been defined and generalized in terms of various objects. The most widely used ones are certain metrics, relations and covers. Covers are versatile tools in topology, while metrics are better adapted to arguments in analysis. However, almost everything can be formulated more simply in terms of relations. Therefore, we adhere to relations. Before describing the main features of our present approach, it seems necessary to make some historical remarks. Uniform spaces in terms of relations were first introduced by Weft [97], and later standardized by Bourbaki [10] with some adjustments. A uniform space in the Weil--Bourbaki sense is an ordered pair X(q/)=(X, ~) consisting of a set X and a nonvoid family q /o f relations U c X • such that

A particular Galois connection between relations and set functions

Abstract Motivated by a recent paper of U. Höhle and T. Kubiak on regular sup-preserving maps, we investigate a particular Galois-type connection between relations on one set X to another Y and functions on the power set P(X) to P(Y ) . Since relations can largely be identified with union-preserving set functions, the results obtained can be used to provide some natural generalizations of most of the former results on relations and relators (families of relations). The results on inverses seem to be the only exceptions.

Mild Continuity Properties of Relations and Relators in Relator Spaces

In this paper, we establish several useful consequences of the following, and some other closely related, basic definitions introduced in some former papers by the first author. A family \( \mathcal{R} \) of relations on one set X to another Y is called a relator on X to Y. Moreover, the ordered pair \( (\,X\,,\,Y \,)(\,\mathcal{R}\,) ={\bigl (\, (\,X\,,\,Y \,),\ \mathcal{R}\,\bigr )} \) is called a relator space. A function \( \square \) of the class of all relator spaces to the class of all relators is called a direct unary operation for relators if, for any relator \( \mathcal{R} \) on X to Y, the value \( \mathcal{R}^{\,\,\square } = \mathcal{R}^{\ \square _{X\,Y }} = \square \,{\bigl ((\,X,\,Y \,)(\,\mathcal{R}\,)\bigr )} \) is also relator on X to Y. If \( (\,X\,,\,Y \,)(\,\mathcal{R}\,) \) and \( (\,Z\,,\,W\,)(\,\mathcal{S}\,) \) are relator spaces and \( \square \) is a direct unary operation for relators, then a pair \( (\,\mathcal{F}\,,\ \mathcal{G}\,) \) of relators \( \mathcal{F} \) on X to Z and \( \mathcal{G} \) on Y to W is called mildly \( \square \)–continuous if, under the elementwise inversion and compositions of relators, we have \( \bigl ((\mathcal{G}^{\,\square }\,)^{-1}\! \circ \,\mathcal{S}^{\,\square }\circ \,\mathcal{F}^{\,\,\square }\,\bigr )^{\square }\subseteq \mathcal{R}^{\ \square \,\square } \).

Basic Tools, Increasing Functions, and Closure Operations in Generalized Ordered Sets

Having in mind Galois connections, we establish several consequences of the following definitions.An ordered pair X( ≤ ) = (X, ≤ ) consisting of a set X and a relation ≤ on X is called a goset (generalized ordered set).For any x ∈ X and \(A \subseteq X\), we write x ∈ ub X (A) if a ≤ x for all a ∈ A, and \(x \in \mathop{\mathrm{int}}\nolimits _{X}(A)\) if \(\mathop{\mathrm{ub}}\nolimits _{X}(x) \subseteq A\), where \(\mathop{\mathrm{ub}}\nolimits _{X}(x) =\mathop{ \mathrm{ub}}\nolimits _{X}{\bigl (\{x\}\bigr )}\).Moreover, for any \(A \subseteq X\), we also write \(A \in \mathcal{U}_{X}\) if \(A \subseteq \mathop{\mathrm{ub}}\nolimits _{X}(A)\), and \(A \in \mathcal{T}_{X}\) if \(A \subseteq \mathop{\mathrm{int}}\nolimits _{X}(A)\). And in particular, \(A \in \mathcal{E}_{X}\) if \(\mathop{\mathrm{int}}\nolimits _{X}(A)\neq \emptyset\) .A function f of one goset X to another Y is called increasing if u ≤ v implies f(u) ≤ f(v) for all u, v ∈ X.In particular, an increasing function \(\varphi\) of X to itself is called a closure operation if \(x \leq \varphi (x)\) and \(\varphi {\bigl (\varphi (x)\bigr )} \leq \varphi (x)\) for all x ∈ X.The results obtained extend and supplement some former results on increasing functions and can be generalized to relator spaces.

A natural Galois connection between generalized norms and metrics

Abstract Having in mind a well-known connection between norms and metrics on vector spaces, for an additively written group X, we establish a natural Galois connection between functions of X to ℝ and X2 to ℝ.

Constructions and Extensions of Free and Controlled Additive Relations

By using several auxiliary results on relations and their intersection convolutions, we give some necessary and sufficient conditions in order that a certain additive partial selection relation Φ of a relation F of one group X to another Y could be extended to a total, additive selection relation Ψ of the relation \(F+\Phi(0)\).

Corelations Are More Powerful Tools than Relations

A subset R of a product set Open image in new window is called a relation on X to Y . While, a function U of one power set \(\mathcal {P}(X)\) to another \(\mathcal {P}(Y)\) is called a corelation on X to Y . Moreover, families \(\mathcal {R}\) and \(\mathcal {U}\) of relations and corelations on X to Y are called relators and corelators on X to Y , respectively.

The Hyers–Ulam and Hahn–Banach Theorems and Some Elementary Operations on Relations Motivated by Their Set-Valued Generalizations

In the first part of this paper, we provide several historical facts on the famous Hyers–Ulam stability theorems, Hahn–Banach extension theorems, and their set-valued generalizations with numerous references.

ON THE EXISTENCE OF NONNEGATIVITY DOMAINS OF SUBSETS OF GROUPS

1. A few basic facts on families of sets and subsets of groups A family A of sets is called chained if for any A, B G A we have either A C B or B C A. The family A is called directed if for any A, B 6 A there exists C € A such that A C C and B C C. A subfamily B of a family of sets A is called bounded above in A if there exists A € A such that B C A for all B € B. Moreover, the family A is called inductive if each chained subfamily of A is bounded above in A. 1991 Mathematics Subject Classification: 06P15.