Sublattices of the direct product (Q998776)

The authors establish an anti-isomorphism between the lattice \(\text{Subd}^{01}(K, M)\) of all \((0, 1)\)-subdirect products of complete lattices \(K\) and \(M\), and the lattice \(\text{Res}(K, M)\) of adjunctions (residuated mappings) from \(K\) to \(M\). They then prove that for every subdirect product \(L\) of two complete lattices \(K\) and \(M\) there exist join-preserving mappings \(\varphi : K \rightarrow M\) and \(\alpha : M \rightarrow K\) and meet-preserving mappings \(\psi : M \rightarrow K\), \(\beta : K \rightarrow M\) such that: \[ \{ (x, y) \mid \varphi(x) \leq y \leq \beta(x) \} = L = \{ (x, y) \mid \alpha(y) \leq x \leq \psi(y) \}. \] This result is used for: {\parindent=7mm \begin{itemize}\item[(1)] obtaining a necessary and sufficient condition for an intersection of two subdirect products of complete lattices to be itself a subdirect product of the lattices; \item[(2)] obtaining a characterization of subdirect products of a complete lattice \(K\) with itself which are symmetric (as binary relations); \item[(3)] characterizing complete tolerances of a complete lattice \(K\); \item[(4)] characterizing complete congruences of a complete lattice \(K\). \end{itemize}} They then prove that the problem of describing all \((0,1)\)-subdirect products of two complete lattices \(K\) and \(M\) reduces to a special case, called ``edge minimal''. Finally, the authors show that the existence of an anti-isomorphism between \(\text{Subd}^{01}(K, M)\) and \(\text{Res}(K, M)\) can also be derived from general results in formal concept analysis.