Existentially Closed Brouwerian Semilattices

Abstract

The variety of Brouwerian semilattices is amalgamable and locally finite, hence by well-known results [Whe76], it has a model completion (whose models are the existentially closed structures). In this paper, we supply for such a model completion a finite and rather simple axiomatization.2010 Mathematics subject classification. Primary 03G25; Secondary 03C10, 06D20.A co-Brouwerian semilattice, CBS for short, is a structure obtained by reversing the order of a Brouwerian semilattice. We will work with CBSes instead of Brouwerian semilattices.Definition 2.1. A poset (P, ≤) is said to be a co-Brouwerian semilattice if it has a least element, which we denote with 0, and for every a, b ∈ P there exists the sup of {a, b}, which we call join of a and b and denote with a ∨ b, and the difference a − b satisfying for every c ∈ PClearly, there is also an alternative equational definition for co-Brouwerian semilattices (which we leave to the reader, because it is dual to the equational definition for Brouwerian semilattices given above).Moreover, we will call co-Heyting algebras the structures obtained reversing the order of Heyting algebras. Obviously any co-Heyting algebra is a CBS.Definition 2.2. Let A, B be co-Brouwerian semilattices. A map f : A → B is a morphism of co-Brouwerian semilattices if it preserves 0, the join and difference of any two elements of A.Notice that such a morphism f is an order preserving map because, for any a, b elements of a co-Brouwerian semilattice, we have a ≤ b iff a ∨ b = b.Definition 2.3. Let L be a CBS. We say that g ∈ L is join-irreducible iff for every n ≥ 0 and b 1 , . . . , b n ∈ L, we have that g ≤ b 1 ∨ . . . ∨ b n implies g ≤ b i for some i = 1, . . . , n.Notice that taking n = 0 we obtain that join-irreducibles are different from 0.Remark 2.4. Let L be a CBS and g ∈ L. Then the following conditions are equivalent:1. g is join-irreducible.2. g = 0 and for any b 1 , b 2 ∈ L we have that g3. For every n ≥ 0 and b 1 , . . . , b n ∈ L we have that g = b 1 ∨ . . . ∨ b n implies g = b i for some i = 1, . . . , n.4. g = 0 and for any b 1 , b 2 ∈ L we have that g = b 1 ∨ b 2 implies g = b 1 or g = b 2 .5. g = 0 and for any a ∈ L we have that g − a = 0 or g − a = g.Proof. The implications 1 ⇔ 2, 3 ⇔ 4 and 1 ⇒ 3 are straightforward. For the remaining ones see Lemma 2.1 in [Köh81].Definition 2.5. Let L be a CBS and a ∈ L.A join-irreducible component of a is a maximal element among the join-irreducibles of L that are smaller than or equal to a.

Admissibility of $Π_2$-Inference Rules: interpolation, model completion, and contact algebras

“…First notice that, as a consequence of (10), if the cardinality of X B0 and of X C is the same, then f is an isomorphism. This is seen as follows: we already observed that condition (10) implies surjectivity and in case of the same finite cardinality surjectivity implies injectivity. Preservation and reflection of the relation follow by stability and (10) again.…”

“…Proof. We need to show that f satisfies condition (10) above iff f ⋆ is an embedding between contact algebras, i.e. iff it satisfies the condition…”

“…where Clop(X 2 ) is the set of clopens of the Stone space X 2 . We tranform condition (11) up to equivalence. First notice that, by the adjunction between direct and inverse image, ( 11) is equivalent to…”

“…We recall that Con ⋆ is the theory of existentially closed contact algebras [14]. The following result (given that Con is locally finite) is folklore (a detailed proof of the analogous statement for Brouwerian semilattices is in the ArXiv version of [11] as [10,Proposition 2.16]). Theorem 6.7.…”

“…10 A binary relation R on a topological space X is said to be point-closed if ∀x ∈ X R[x] is closed in X. A closed relation in a compact Hausdorff space maps closed sets to closed sets via R[−], hence it is point-closed.…”