Equational theories as congruences of enriched monoids (Q934894)

For a given equational theory \(\Sigma\), the set \(L(\Sigma)\) of all equational theories containing \(\Sigma\), forms a lattice, called an equational theories lattice. Mal'tsev's problem, which lattices can be represented as equational theories lattices, is still unsolved. \textit{J.~Ježek} [Algebra Univers. 6, 147--158 (1976; Zbl 0354.08007)] proved that each algebraic lattice with countably many compact elements is isomorphic to an interval of some equational theories lattice. \textit{D.~Pigozzi} and \textit{G.~Tardos} [manuscript, 1999] showed that for every algebraic lattice \(L\), the ordinal sum \(L+1\) is representable as an equational theories lattice. \textit{N.~Newrly} [Algebra Univers. 30, No. 2, 217--220 (1993; Zbl 0789.08006)] studied representations of equational theories lattices by congruence lattices of monoids enriched by one unary operation. The author of the present paper defines an algebra \(\mathcal M=(M,\cdot,\Phi, \Phi_0,1)\) of type (2,1,1,0) to be an Et-monoid, if \((M,\cdot,1)\) is a monoid and if certain further conditions for \(\mathcal M\) are satisfied. The main result is the following, constructively proved, theorem: A lattice \(L\) is an equational theories lattice if and only if \(L\) is isomorphic to the congruence lattice of some Et-monoid.