Varieties of semigroups with non-trivial identities on subsemigroup lattices. (Q1771859)

It is firstly shown that for a variety \(\mathcal V\) of groups, if the lattice \(\text{Sub\,}{\mathcal V}\) satisfies a non-trivial identity then \(\mathcal V\) is soluble. Then the author uses his previous result that every lattice can be embedded into the subsemigroup lattice of a suitable semilattice and also into the subsemigroup lattice of a suitable commutative nilsemigroup of index two. Applying these results, he characterizes semigroup varieties \(\mathcal V\) such that for \(S\in{\mathcal V}\), \(\text{Sub\,}S\) satisfies a non-trivial lattice identity by means of several conditions (e.g. the free lattice of countable rank is not embeddable into \(\text{Sub\,}S\) for each \(S\in{\mathcal V}\)).