Lattices of algebraic subsets

Abstract

This survey article tackles different aspects of lattices of algebraic subsets, with the emphasis on the following: the theory of quasivarieties, general lattice theory and the theory of closure spaces with the anti-exchange axiom.

On the prevariety of perfect lattices

“…We show in this paper that every lower bounded lattice in the class of lattices of suborders is perfect (Corollary 5.2). In general, it remains unknown whether every lower bounded lattice is perfect (see Problem 2 in [2]). …”

“…Notice that the relations between the elements mimic those of the corresponding elements of the semilattice from Lemma 5.8, case (2), in the sense that a k+1 ≤ a k ∨ b k+1 holds in O(P, ≤). Define the suborders X = k≥2 b k ∨ b ω , Y = k≥1 a 2k ∨ k≥1 b 2k , and Z = k≥0 a 2k+1 ∨ k≥1 b 2k+1 .…”

“…It is easy to check that φ ≤ ψ iff every ψ-closed set is φ-closed. In particular, the algebraic closure operators form a complete lattice which is dually isomorphic to the lattice of algebraic subsets of B(X), see also [2].…”

The Jónsson-Kiefer Property

“…It is also true that if R is distributive and A is Scott continuous, then Sp(A, R) is dually Scott continuous (see, for example, Adaricheva [2] for the case of R = id).…”

“…Statement (1) is straightforward to verify. For (2), assume that {p, p , q} ⊆ Π, p = p , and p ∩ p ⊆ q. Let s ∈ S ω be the least member of p ∩ p and s ≺ σ be the subcover of s that lies off q.…”