Generation and decidability for periodic l-pregroups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929268623147008 |
|---|---|
| author | Galatos, Nikolaos Gallardo, Isis A. |
| author_facet | Galatos, Nikolaos Gallardo, Isis A. |
| contents | In [11] it is shown that the variety $\mathsf{DLP}$ of distributive l-pregroups is generated by a single algebra, the functional algebra $\mathbf{F}(Z)$ over the integers. Here, we show that $\mathsf{DLP}$ is equal to the join of its subvarieties $\mathsf{LPn}$, for $n\in\mathbb{Z}$, consisting of n-periodic l-pregroups. We also prove that every algebra in $\mathsf{LPn}$ embeds into the subalgebra $\mathbf{F}_n(Ω)$ of n-periodic elements of $\mathbf{F}(Ω)$, for some integral chain $Ω$; we use this representation to show that for every n, the variety $\mathsf{LPn}$ is generated by the single algebra $\mathbf{F}_n(\mathbb{Q}\overrightarrow{\times}\mathbb{Z})$, noting that the chain $\mathbb{Q}\overrightarrow{\times}\mathbb{Z}$ is independent of n. We further establish a second representation theorem: every algebra in $\mathsf{LPn}$ embeds into the wreath product of an l-group and $\mathbf{F}_n(\mathbb{Z})$, showcasing the prominent role of the simple n-periodic l-pregroup $\mathbf{F}_n(\mathbb{Z})$. Moreover, we prove that the join of the varieties $V(\mathbf{F}_n(\mathbb{Z}))$ is also equal to $\mathsf{DLP}$, hence equal to the join of the varieties $\mathsf{LPn}$, even though $\mathsf{V}(\mathbf{F}_n(\mathbb{Z}))$ is not equal to \mathsf{LPn} for every single n. In this sense, $\mathsf{DLP}$ has two different well-behaved approximations. We further prove that, for every n, the equational theory of $\mathbf{F}_n(\mathbb{Z})$ is decidable and, using the wreath product decomposition, we show that the equational theory of $\mathsf{LPn}$ is decidable, as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05099 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generation and decidability for periodic l-pregroups Galatos, Nikolaos Gallardo, Isis A. Logic In [11] it is shown that the variety $\mathsf{DLP}$ of distributive l-pregroups is generated by a single algebra, the functional algebra $\mathbf{F}(Z)$ over the integers. Here, we show that $\mathsf{DLP}$ is equal to the join of its subvarieties $\mathsf{LPn}$, for $n\in\mathbb{Z}$, consisting of n-periodic l-pregroups. We also prove that every algebra in $\mathsf{LPn}$ embeds into the subalgebra $\mathbf{F}_n(Ω)$ of n-periodic elements of $\mathbf{F}(Ω)$, for some integral chain $Ω$; we use this representation to show that for every n, the variety $\mathsf{LPn}$ is generated by the single algebra $\mathbf{F}_n(\mathbb{Q}\overrightarrow{\times}\mathbb{Z})$, noting that the chain $\mathbb{Q}\overrightarrow{\times}\mathbb{Z}$ is independent of n. We further establish a second representation theorem: every algebra in $\mathsf{LPn}$ embeds into the wreath product of an l-group and $\mathbf{F}_n(\mathbb{Z})$, showcasing the prominent role of the simple n-periodic l-pregroup $\mathbf{F}_n(\mathbb{Z})$. Moreover, we prove that the join of the varieties $V(\mathbf{F}_n(\mathbb{Z}))$ is also equal to $\mathsf{DLP}$, hence equal to the join of the varieties $\mathsf{LPn}$, even though $\mathsf{V}(\mathbf{F}_n(\mathbb{Z}))$ is not equal to \mathsf{LPn} for every single n. In this sense, $\mathsf{DLP}$ has two different well-behaved approximations. We further prove that, for every n, the equational theory of $\mathbf{F}_n(\mathbb{Z})$ is decidable and, using the wreath product decomposition, we show that the equational theory of $\mathsf{LPn}$ is decidable, as well. |
| title | Generation and decidability for periodic l-pregroups |
| topic | Logic |
| url | https://arxiv.org/abs/2403.05099 |