Axiomatizing small varieties of periodic l-pregroups

Abstract

We provide an axiomatization for the variety generated by the $n$-periodic l-pregroup ${\mathbf{F}}_n(\mathbb{Z})$, for every $n\in {\mathbb{Z}}^{+}$, as well as for all possible joins of such varieties; the finite joins form an ideal in the subvariety lattice of l-pregroups and we describe fully its lattice structure. On the way, we characterize all finitely subdirectly irreducible (FSI) algebras in the variety generated by ${\mathbf{F}}_n(\mathbb{Z})$ as the $n$-periodic l-pregroups that have a totally ordered group skeleton (and are not trivial). The finitely generated FSIs that are not l-groups are further characterized as lexicographic products of a (finitely generated) totally ordered abelian l-group and ${\mathbf{F}}_k(\mathbb{Z})$, where $k\mid n$.

54 pages, 5 figures