Mild Continuity Properties of Relations and Relators in Relator Spaces

Árpád Száz () and Amr Zakaria ()
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Árpád Száz: University of Debrecen, Institute of Mathematics
Amr Zakaria: Ain Shams University, Department of Mathematics

A chapter in Essays in Mathematics and its Applications, 2016, pp 439-511 from Springer

Abstract: Abstract 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 $$ \bigl ((\mathcal{G}^{\,\square }\,)^{-1}\! \circ \,\mathcal{S}^{\,\square }\circ \,\mathcal{F}^{\,\,\square }\,\bigr )^{\square }\subseteq \mathcal{R}^{\ \square \,\square } $$ .

Keywords: Relative Spacing; Mild Continuity; Compatible Composition; Detailed Reformulations; Proximal Interior (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-31338-2_16

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DOI: 10.1007/978-3-319-31338-2_16

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