The lattice of varieties of semigroups with completely regular square. II

Abstract

We solve (modulo groups) the word problem for free semigroup satisfying $xy=(xy{)}^n$. This enables us to show that the variety ${\mathcal{SC}\mathcal{R}}_n$ given by this identity is not equal to the join of the varieties ${\mathcal{C}\mathcal{R}}_n$ and $\mathcal{S}\mathcal{I}$ defined by the identities $x=x^n$ and $xy=(xy{)}^2$ respectively. Therefore the result of (M.V.Volkov, T.A.Ershova, The lattice of varieties of semigroup with completely regular square, Monash Conference on Semigroup Theory in Honour of G. B. Preston, World Scientific, Singapore, 1991, 306-322) that the lattice $L({\mathcal{SC}\mathcal{R}}_n)$ is modular does not follow from the earlier results concerning $L({\mathcal{C}\mathcal{R}}_n)$ and $L(\mathcal{S}\mathcal{I})$. However the variety ${\mathcal{S}\mathcal{O}}_n$ consisting of all semigroups from ${\mathcal{SC}\mathcal{R}}_n$ whose idempotents form a subsemigroup turns out to be equal to the join of the varieties ${\mathcal{O}}_n$ of all orthogroups and $\mathcal{S}\mathcal{I}$. We deduce from this a description of the lattice $L({\mathcal{S}\mathcal{O}}_n)$ modulo groups.

arXiv commentThis paper was written in 1991 as a continuation of (M.V.Volkov, T.A.Ershova, The lattice of varieties of semigroup with completely regular square, Monash Conference on Semigroup Theory in Honour of G. B. Preston, World Scientific, Singapore, 1991, 306-322), but it was never submitted for publication. In the present version, a few typos have been corrected