Brouwerian semilattices

Abstract:

Let ${\mathbf {P}}$ be the category whose objects are posets and whose morphisms are partial mappings $\alpha :P \to Q$ satisfying (i) $\forall p, q \in \operatorname {dom} \alpha [p < q \Rightarrow \alpha (p) < \alpha (q)]$ and (ii) $\forall p \in \operatorname {dom} \alpha \forall q \in Q[q < \alpha (p) \Rightarrow \exists r \in \operatorname {dom} \alpha [r < p\& \alpha (r) = q]]$. The full subcategory ${{\mathbf {P}}_f}$ of ${\mathbf {P}}$ consisting of all finite posets is shown to be dually equivalent to the category of finite Brouwerian semilattices and homomorphisms. Under this duality a finite Brouwerian semilattice $\underline A$ corresponds with $M(\underline A )$, the poset of all meet-irreducible elements of $\underline A$. The product (in ${{\mathbf {P}}_f}$) of $n$ copies $(n \in \mathbb {N})$ of a one-element poset is constructed; in view of the duality this product is isomorphic to the poset of meet-irreducible elements of the free Brouwerian semilattice on $n$ generators. If ${\mathbf {V}}$ is a variety of Brouwerian semilattices and if $\underline A$ is a Brouwerian semilattice, then $\underline A$ is ${\mathbf {V}}$-critical if all proper subalgebras of $\underline A$ belong to ${\mathbf {V}}$ but not $\underline A$. It is shown that a variety ${\mathbf {V}}$ of Brouwerian semilattices has a finite equational base if and only if there are up to isomorphism only finitely many ${\mathbf {V}}$-critical Brouwerian semilattices. This is used to show that a variety generated by a finite Brouwerian semilattice as well as the join of two finitely based varieties is finitely based. A new example of a variety without a finite equational base is exhibited.