Esakia order-compactifications and locally Esakia spaces

TL;DR

This paper introduces Esakia order-compactifications, characterizes them via rings of upsets, and explores locally Esakia spaces, establishing functorial properties and analogs of classical theorems in order-compactification theory.

Contribution

It develops the theory of Esakia order-compactifications, provides a characterization using rings of upsets, and introduces locally Esakia spaces with functorial properties.

Findings

Esakia order-compactifications characterized by rings of upsets.

Largest Esakia order-compactification is functorial.

Established analogs of Dwinger's and Banaschewski's theorems.

Abstract

We introduce Esakia order-compactifications and study how they fit in the general theory of Priestley order-compactifications. We provide an analog of Dwinger's theorem by characterizing Esakia order-compactifications by means of special rings of upsets. These considerations naturally lead to the notion of a locally Esakia space, for which we prove that taking the largest Esakia order-compacification is functorial, thus obtaining an analog of Banaschewski's theorem.