Topological Representation of Precontact Algebras and a Connected Version of the Stone Duality Theorem -- I

TL;DR

This paper introduces new topological structures called 2-precontact and 2-contact spaces to represent precontact and contact algebras, leading to connected versions of the Stone Duality Theorem and new representation theorems.

Contribution

It develops novel topological representations for precontact and contact algebras, including connected variants of classical duality theorems and introduces Stone adjacency spaces for algebraic correspondence.

Findings

Established bijective correspondences between precontact/contact algebras and topological spaces.

Derived new connected versions of the Stone Duality Theorem for Boolean algebras.

Proved a representation theorem linking precontact algebras with Stone adjacency spaces.

Abstract

The notions of a {\em 2-precontact space}\/ and a {\em 2-contact space}\/ are introduced. Using them, new representation theorems for precontact and contact algebras are proved. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. As applications of the obtained results, we get new connected versions of the Stone Duality Theorems for Boolean algebras and for complete Boolean algebras, as well as a Smirnov-type theorem for a kind of compact $T_0$-extensions of compact Hausdorff extremally disconnected spaces. We also introduce the notion of a {\em Stone adjacency space}\/ and using it, we prove another representation theorem for precontact algebras. We even obtain a bijective correspondence between the class of all, up to isomorphism, precontact algebras and the class of all, up to isomorphism, Stone adjacency spaces.