Axiomatizing small varieties of periodic l-pregroups
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915796652916736 |
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| author | Galatos, Nikolaos Santschi, Simon |
| author_facet | Galatos, Nikolaos Santschi, Simon |
| contents | 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 $k \mid n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Axiomatizing small varieties of periodic l-pregroups Galatos, Nikolaos Santschi, Simon Rings and Algebras Logic 06F05, 06F15, 03C05 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 $k \mid n$. |
| title | Axiomatizing small varieties of periodic l-pregroups |
| topic | Rings and Algebras Logic 06F05, 06F15, 03C05 |
| url | https://arxiv.org/abs/2503.18660 |