## Relationships between inclusions for relations and inequalities for corelations.(English)Zbl 1424.06009

Summary: Let $$X$$ and $$Y$$ be quite arbitrary sets. Then, a function $$U$$ on the power set $$\mathcal{P}(X)$$ to $$\mathcal{P}(Y)$$ will be called a corelation on $$X$$ to $$Y$$. Thus, complementation and closure (interior) operations on $$X$$ are corelations on $$X$$.
Moreover, for any two corelations $$U$$ and $$V$$ on $$X$$ to $$Y$$, we shall write $$U\le V$$ if $$U(A)\subseteq V(A)$$ for all $$A\subseteq X$$. Thus, the family of all corelations on $$X$$ to $$Y$$ also forms a complete poset (partially ordered set).
Formerly, we have established a partial Galois connection $$(\triangleright,\triangleleft)$$, between relations and corelations. Now, by using this, we shall establish some further relationships between inclusions for relations and inequalities for corelations.
For instance, for some very particular corelations $$U$$ and $$V$$ on $$X$$ to $$Y$$, with $$U^{\triangleleft}\le V^{\triangleleft}$$, we shall prove the existence of an union-preserving corelation $$\Phi$$ on $$X$$ to $$Y$$ which separates $$U$$ and $$V$$ in the sense that $$U\le \Phi\le V$$ .

### MSC:

 06A15 Galois correspondences, closure operators (in relation to ordered sets) 54C60 Set-valued maps in general topology

### Keywords:

relations; setfunctions; Galois connections