Inclusions for Compositions and Box Products of Relations

\'Arp\'ad Sz\'az

Motivated by some important classes and continuities of relations, we give some intrinsic characterizations of the relational inclusions \[ G^{-1}\circ S\circ F\subset R\qquad \text{and} \qquad R\subset G^{-1}\circ S\circ F. \] Our main tool is the box product $F\boxtimes G$ which has the crucial property that \[ G^{-1}\circ S\circ F=(F^{-1}\boxtimes G^{-1})[S]=(F\boxtimes G)^{-1}[S]=\operatorname{cl}_{F\boxtimes G}(S). \] For any fixed relations $F$, $G$, and $R$, by considering the extremal relation \[ S_({F,G,R})=\operatorname{int}_{(F\boxtimes G)^{-1}}(R)=\{(y,w):\quad(F\boxtimes G)^{-1}(y,w)\subset R\}, \] we give some easily applicable necessary and sufficient conditions in order that the equality $R=G^{-1}\circ S\circ F$ could hold for some relation $S$. The results obtained extend and unify several former results on transitive, idempotent, non-mingled-valued, regular, normal, and conjugative relations, for instance. Moreover, they can also be well used to briefly treat proper and uniform, mild, upper, and lower continuity properties of a pair $(F,G)$ of relations on one relator space $(X,Z)(R)$ to another $(Y,W)(\mathcal S)$.