Generation and decidability for periodic l-pregroups

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 $\mathbf{F}_n(\Omega)$ of n-periodic elements of $\mathbf{F}(\Omega)$, for some integral chain $\Omega$; 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.