A family of varieties of pseudosemilattices. (Q896227)

A regular semigroup \(S\) is locally inverse if \(eSe\) is an inverse subsemigroup for every idempotent \(e\in S\). A binary operation may be defined on the set \(E\) of idempotents of \(S\) by \(e\wedge f=eV(fe)f\), where \(V(fe)\) is the set of inverses of \(fe\). The pseudosemilattices are the algebras \((E,\wedge)\) that characterize the outcome abstractly, a set of defining identities having been given by \textit{K. S. S. Nambooripad} [Simon Stevin 55, 103-110 (1981; Zbl 0467.06003)]. \textit{K. Auinger} and \textit{L. Oliveira} [Stud. Sci. Math. Hung. 50, No. 2, 207-241 (2013; Zbl 1299.06034)] found a sequence of identities defining the subvariety of strict pseudosemilattices, corresponding to the class of strict regular semigroups, based on the notion of 2-content, which is a little too complicated to state here. The \(n^{th}\) identity in the sequence has a particular 2-content \(D_n\). The purpose of this paper is to use graphical methods to determine all the varieties defined by identities with a specific 2-content \(D_n\). In fact, each is definable by a single such identity. The paper concludes with an illuminating diagrammatic representation of the set of such varieties.