Generation and decidability for periodic l-pregroups

In [11] it is shown that the variety 𝖣𝖫𝖯 of distributive l-pregroups is generated by a single algebra, the functional algebra 𝐅(Z) over the integers. Here, we show that 𝖣𝖫𝖯 is equal to the join of its subvarieties 𝖫𝖯𝗇, for n∈ℤ, consisting of n-periodic l-pregroups. We also prove that every algebra in 𝖫𝖯𝗇 embeds into the subalgebra 𝐅_n(Ω) of n-periodic elements of 𝐅(Ω), for some integral chain Ω; we use this representation to show that for every n, the variety 𝖫𝖯𝗇 is generated by the single algebra 𝐅_n(ℚ×ℤ), noting that the chain ℚ×ℤ is independent of n. We further establish a second representation theorem: every algebra in 𝖫𝖯𝗇 embeds into the wreath product of an l-group and 𝐅_n(ℤ), showcasing the prominent role of the simple n-periodic l-pregroup 𝐅_n(ℤ). Moreover, we prove that the join of the varieties V(𝐅_n(ℤ)) is also equal to 𝖣𝖫𝖯, hence equal to the join of the varieties 𝖫𝖯𝗇, even though 𝖵(𝐅_n(ℤ)) is not equal to 𝖫𝖯𝗇 for every single n. In this sense, 𝖣𝖫𝖯 has two different well-behaved approximations. We further prove that, for every n, the equational theory of 𝐅_n(ℤ) is decidable and, using the wreath product decomposition, we show that the equational theory of 𝖫𝖯𝗇 is decidable, as well.