A note on lattice congruence regularization. (Q1771867)

An algebra has regular congruences if each congruence is uniquely determined by each of its congruence classes. An extension \(K\) of a lattice \(L\) is called congruence-preserving extension if each congruence on \(L\) has a unique extension to a congruence on \(K\). G. Grätzer and E. T. Schmidt proved that any lattice \(L\) has a congruence-preserving extension \(K\) with regular congruences. They also posed the following problem: Is it true that, given a lattice \(L\), there is a congruence-preserving embedding \(\varphi : L \rightarrow M_3 (L)\) such that for any \(a,b,c \in L\) with \(a<b\) there are \(d,e\in M_3(L)\) with \(d<\varphi (c) < e\) and with \(\Theta (d,e) = \Theta ( \varphi (a), \varphi (b) )\)? The author constructs an example that answers this question in the negative.