Some infinite chains in the lattice of varieties of inverse semigroups (Q909038)

The relation \(\vee\) defined on the lattice \({\mathcal L}({\mathcal I})\) of varieties of inverse semigroups by \({\mathcal U}\vee {\mathcal V}\) if and only if \({\mathcal U}\cap {\mathcal G}={\mathcal V}\cap {\mathcal G}\) and \({\mathcal U}\vee {\mathcal G}={\mathcal V}\vee {\mathcal G}\), where \({\mathcal G}\) is the variety of groups, is a congruence. It is known that varieties belonging to the first three layers of \({\mathcal L}({\mathcal I})\) (those varieties belonging to the lattice \({\mathcal L}({\mathcal S}{\mathcal I})\) of varieties of strict inverse semigroups) possess trivial \(\vee\)-classes and that there exist non-trivial \(\vee\)- classes in the next layer of \({\mathcal L}({\mathcal I})\). We show that there are infinitely many \(\vee\)-classes in the fourth layer of \({\mathcal L}({\mathcal I})\), and also higher up in \({\mathcal L}({\mathcal I})\), that in fact contain an infinite descending chain of varieties. To find these chains we first construct a collection of semigroups in \({\mathcal B}^ 1\), the variety generated by the five element combinatorial Brandt semigroup with an identity adjoined. By considering wreath products of abelian groups and these semigroups from \({\mathcal B}^ 1\), we obtain an infinite descending chain in the \(\vee\)-class of \({\mathcal U}\vee {\mathcal B}^ 1\), for every nontrivial abelian group variety \({\mathcal U}\).