Congruences, equational theories and lattice representations (Q1922118)

This paper is an interesting survey, concerning the representation of lattices as lattices of congruences and as lattices of equational theories. For an algebra \(A\), \(\text{Con}(A)\) is the lattice of congruence relations, \(\text{Eq}(A)\) the lattice of equivalence relations on \(A\). Every algebraic lattice \(L\) has a representation as a type 3 complete sublattice of some \(\text{Eq}(A)\). If \(L\) is in addition modular, then \(L\) has a representation as a type 2 complete sublattice of some \(\text{Eq}(A)\). \(L\) is an algebraic lattice iff there exists an algebra \(A\) having type 3 congruence joins such that \(\text{Con}(A)\) is isomorphic to \(L\). Also, \(A\) can be chosen such that \(\Aut(A)\) is isomorphic to any given group. If \(D\) is a finite distributive lattice, then there is a finite groupoid \(A\) with \(\text{Con}(A)\) isomorphic to \(D\). If \(D\) is an algebraic distributive lattice then there is a semigroup \(A\) with \(\text{Con}(A)\) isomorphic to \(D\). Further definitions are needed, concerning results on similarity type, compact 1, lattices of equational theories. Some problems and conjectures are stated.