Brouwerian semilattices

Abstract

Abstract. Let P be the category whose objects are posets and whose morphisms are partial mappings a: P-> Q satisfying (i) V p,q e dom a [p < q =* a(p) < a(q)] and The full subcategory Py of 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 A corresponds with M(A), the poset of all meet-irreducible elements of A. The product (in P,) of n copies (n E 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 V is a variety of Brouwerian semilattices and if A is a Brouwerian semilattice, then A is V-critical if all proper subalgebras of A belong to V but not A. It is shown that a variety V of Brouwerian semilattices has a finite equational base if and only if there are up to isomorphism only finitely many 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. 0. Introduction. The interest in Brouwerian semilattices has been motivated mainly from two seemingly different directions. First there is intuitionistic propositional logic: The Lindenbaum algebra of its fragment £/c, which consists of formulas containing only implication and conjunction as logical connectives, is a free Brouwerian semilattice on a countable number of generators. Thus a formula is a tautology relative to £/c if and only if it is valid in every Brouwerian semilattice. In addition the validity relation between formulas and Brouwerian semilattices sets up a Galois connection between extensions of £/c and subvarieties of the variety of Brouwerian semilattices. Consequently algebraic methods can be successfully employed to deal with problems of a logical nature. Examples of this approach are the papers [31] and [52]; it must be said, however, that historically it was mostly full intuitionistic propositional logic (with disjunction and possibly negation) that was studied and thus on the algebraic side more interest was created in Brouwerian lattices and Heyting algebras.On the other hand the main pioneer in the study of Brouwerian semilattices, Nemitz, considered them as algebraic objects in their own right. In [34] he initiated

On the free implicative semilattice extension of a Hilbert algebra

“…An implicative semilattice , also known as Brouwerian semilattice 11 and as Hertz algebra 14, is a tuple 〈 L , ∧, →, 1〉 where 〈 L , ∧, 1〉 is a meet‐semilattice with top 1 and for every a ∈ L the map a → ( − ): L → L is a right adjoint to the map a ∧( − ): L → L , that is for every a , b , c ∈ L , For every implicative semilattice L , the meet‐semilattice 〈 L , ∧〉 is distributive. Implicative semilattices form a variety; equational axiomatizations, as well as other informations, can be found in 6, 11, 13, 7.…”

“…In 14, Porta introduces the notion of the free Hertz algebra extension of a Hilbert algebra. Hertz algebras in the literature are also known as Brouwerian semilattices 11 and as implicative semilattices 13. In the paper, we shall use the later terminology.…”

Subdirectly irreducible sectionally pseudocomplemented semilattices

“…[8], [7]), filters of relatively pseudocomplemented semilattices are in a one-to-one correspondence with their congruence relations. More precisely, given a filter F of (S, ∧, * , 1), the relation Θ F defined via…”

“…[7], [8]) that a relatively pseudocomplemented semilattice (S, ∧, * , 1) is subdirectly irreducible if and only if it has a smallest non-trivial filter; in other words, the set S \ {1} has a greatest element. This easily follows from the fact that filters agree with congruence kernels.…”

“…Relatively pseudocomplemented semilattices appear in the literature also under the name Brouwerian semilattices or implicative semilattices, respectively (see [7], [8]). For every a ∈ S, we call the principal filter [a) = {x ∈ S : x a} a section of S. It is easy to see that if (S, ∧, * , 1) is a relatively pseudocomplemented semilattice then for any a ∈ S and x ∈ [a), x a := x * a is the pseudocomplement of x in We know that a lattice is relatively complemented if every interval is a complemented lattice.…”