Disjunctions in closure spaces

Abstract

The main result of this paper is the following theorem: a closure space X has an 〈α, δ, Q〉-regular base of the power $\mathfrak{n}$ iff X is Q-embeddable in $B_{\alpha ,\delta }^\mathfrak{n}$ It is a generalization of the following theorems:

• (i) Stone representation theorem for distributive lattices (α = 0, δ = ω, Q = ω),

• (ii) universality of the Alexandroff's cube for T ₀-topological spaces (α = ω, δ = ∞, Q = 0),

• (iii) universality of the closure space of filters in the lattice of all subsets for 〈α, δ〉-closure spaces (Q = 0).

By this theorem we obtain some characterizations of the closure space $F_\mathfrak{m}$ given by the consequence operator for the classical propositional calculus over a formalized language of the zero order with the set of propositional variables of the power $\mathfrak{m}$. In particular we prove that a countable closure space X is embeddable with finite disjunctions preserved into F _ω iff X is a consistent closure space satisfying the compactness theorem and X contains a 〈0, ω〉-base consisting of ω-prime sets.

This paper is a continuation of [7], [2] and [3].