On the lattice of varieties of involution semigroups (Q5936174)

A unary operator * on a semigroup \(S\) is called an involution if \(x^{**}=x\), \((xy)^*=y^*x^*\) for all \(x,y\in S\). Two sublattices of the lattice \(L({\mathcal S}^*)\) of varieties of involution semigroups are described, each generated by atoms of the lattice \(L({\mathcal S}^*)\). The first sublattice contains 18 elements and is generated by the 4 atoms of \(L({\mathcal S}^*)\) which are the varieties generated by, respectively, a 4-element rectangular band with nontrivial involution, a 2-element semilattice with trivial involution, a 3-element semilattice with nontrivial involution, and a 2-element zero semigroup with trivial involution. The second sublattice is generated by the remaining atoms and is isomorphic to the lattice of finite subsets of a countably infinite set. For any \(n\geq 1\), let \({\mathcal A}_n\) [\({\mathcal A}^{id}_n\)] be the variety consisting of the Abelian groups of exponent dividing \(n\) where \(x^*=x^{-1}\) [\(x^*=x\)]. It is shown that the second sublattice consists of the trivial variety and the varieties \({\mathcal A}_m\), \({\mathcal A}^{id}_n\) and \({\mathcal A}_m\vee{\mathcal A}^{id}_n\), where \(m,n\geq 2\) are square-free positive integers.