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