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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Brouwerian semilattices

Author: Peter Köhler
Journal: Trans. Amer. Math. Soc. 268 (1981), 103-126
MSC: Primary 06A12; Secondary 03G10
MathSciNet review: 628448
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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.

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Keywords: Brouwerian semilattice, duality, finitely based variety
Article copyright: © Copyright 1981 American Mathematical Society

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