Lattices of equational theories are congruence lattices of monoids with one additional unary operation
In Stud. Sci Math. Hung. 26, 53-62 (1991; Zbl 0645.08004), the reviewer proved that the class of lattices of equational theories of unary algebras including non-regular equational theories is the class of congruence lattices of monoids with left (or right) zero. Let \(\mathcal M\) be a monoid, \(\langle A,\circ\rangle\) be the monoid obtained from \(\mathcal M\) by means of free addition of left zero \(\omega\), \(h\) be a unary operation on \(A\) such that \(h(a)= h(a\omega)= a\) for any \(a\) from \(\mathcal M\). From the results of the mentioned paper, it follows immediately that the class of congruence lattices of all such algebras \({\mathcal M}(h)=\langle A,\circ,h\rangle\) is the class of lattices of equational theories of unary algebras. In the reviewed paper it is proved that the class of lattices of equational theories of universal algebras is included in the class of congruence lattices of monoids with one additional unary operation.
scientific article; zbMATH DE number 203149