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Upper and Lower Bounds in Relator Spaces

Száz, Árpád — 2003

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 06A06, 54E15 An ordered pair X(R) = ( X, R ) consisting of a nonvoid set X and a nonvoid family R of binary relations on X is called a relator space. Relator spaces are straightforward generalizations not only of uniform spaces, but also of ordered sets. Therefore, in a relator space we can naturally define not only some topological notions, but also some order theoretic ones. It turns out that these two, apparently quite different, types of notions...

The intersection convolution of relations and the Hahn-Banach type theorems

Árpád Száz — 1998

Annales Polonici Mathematici

By introducing the intersection convolution of relations, we prove a natural generalization of an extension theorem of B. Rodrí guez-Salinas and L. Bou on linear selections which is already a substantial generalization of the classical Hahn-Banach theorems. In particular, we give a simple neccesary and sufficient condition in terms of the intersection convolution of a homogeneous relation and its partial linear selections in order that every partial linear selection of this relation can have an...

Supremum properties of Galois-type connections

Árpád Száz — 2006

Commentationes Mathematicae Universitatis Carolinae

In a former paper, motivated by a recent theory of relators (families of relations), we have investigated increasingly regular and normal functions of one preordered set into another instead of Galois connections and residuated mappings of partially ordered sets. A function f of one preordered set X into another Y has been called (1) increasingly   g -normal, for some function g of Y into X , if for any x ∈ X and y ∈ Y we have f ( x ) ≤ y if and only if x ≤ g ( y ) ; (2) increasingly ϕ -regular, for some function ϕ of X into itself,...

Linear extensions of relations between vector spaces

Árpád Száz — 2003

Commentationes Mathematicae Universitatis Carolinae

Let X and Y be vector spaces over the same field K . Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation F of X into Y is called linear if λ F ( x ) ⊂ F ( λ x ) and F ( x ) + F ( y ) ⊂ F ( x + y ) for all λ ∈ K ∖ { 0 } and x , y ∈ X . After improving and supplementing some former results on linear relations, we show that a relation Φ of a linearly independent subset E of X into Y can be extended to a linear relation F of X into Y if and only if there exists a linear subspace Z of Y such that Φ ( e ) ∈ Y | Z for all e ∈ E . Moreover, if E generates...

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