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**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 \(U\le \Phi\le V\) .

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 \(U\le \Phi\le V\) .