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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Brouwerian semilattices
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by Peter Köhler PDF
Trans. Amer. Math. Soc. 268 (1981), 103-126 Request permission

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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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 268 (1981), 103-126
  • MSC: Primary 06A12; Secondary 03G10
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0628448-3
  • MathSciNet review: 628448