On the variety of strict pseudosemilattices

TL;DR

This paper introduces a finite bipartite graph model for free pseudosemilattices to analyze the variety SPS, revealing its complex algebraic structure and properties such as non-finite basis and lack of covers.

Contribution

It provides a new graph-based model for free pseudosemilattices and establishes several fundamental properties of the variety SPS, including its identity basis and non-finite basis nature.

Findings

SPS has an explicit identity basis.

SPS is inherently non-finitely based.

SPS has no covers and is meet-prime in the lattice.

Abstract

A new model, in terms of finite bipartite graphs, of the free pseudosemilattice is presented. This will then be used to obtain several results about the variety SPS of all strict pseudosemilattices: (i) an identity basis for SPS is found, (ii) SPS is shown to be inherently non-finitely based, (iii) SPS is shown to have no irredundant identity basis, and (iv) SPS is shown to have no covers and to be meet-prime in the lattice of all varieties of pseudosemilattices. Some applications to e-varieties of locally inverse semigroups are also derived.