Subdirect products of hereditary congruence lattices (Q2577749)

A finite lattice \({\mathbf L}\) is called representable if it is isomorphic to the congruence lattice \(\text{Con\,}{\mathbf A}\) of a finite universal algebra \({\mathbf A}= (A, F)\). The finite lattice representation problem is the question whether the following statement is true or not: Any finite lattice is representable. Over the last few decades this problem and related questions have been of considerable interest in lattice theory and universal algebra. In the present paper, it is proved that finite lattices \({\mathbf L}\) with special properties are representable. Some of the given representations, so-called hereditary (resp. power-hereditary) representations, are strong in the following sense: If \({\mathbf L}\) is isomorphic to \(\text{Con\,}{\mathbf A}\), then every 0-1 sublattice of \({\mathbf L}\) (resp. \({\mathbf L}^n\), for all positive integers \(n\)) is isomorphic to \(\text{Con}(A, F')\) (resp. to \(\text{Con}(A^n, F')\)) with some suitable family \(F'\) of operations. In particular, the author obtains the following results: (i) If \(\text{Con\,}{\mathbf A}\) is distributive and \({\mathbf B}= (B, G)\) is any universal algebra, then every subdirect product of \(\text{Con\,}{\mathbf A}\) and \(\text{Con\,}{\mathbf B}\) is representable as \(\text{Con}(A\times B, H)\) with some suitable family \(H\) of operations. If, furthermore, \(\text{Con\,}{\mathbf B}\) is power-hereditary, then \(\text{Con\,}{\mathbf A}\times\text{Con\,}{\mathbf B}\) is power-hereditary, too. (ii) Every representation of \({\mathbf N}_5\) (the five-element non-modular lattice) is power-hereditary. (iii) If \(\text{Con\,}{\mathbf A}\) is isomorphic to \({\mathbf N}_5\) and \(\text{Con\,}{\mathbf B}\) is modular, then every subdirect product of \(\text{Con\,}{\mathbf A}\) and \(\text{Con\,}{\mathbf B}\) is representable as in (i).