The Funayama envelope as the $T_D$-hull of a frame

TL;DR

This paper explores the Funayama envelope as the $T_D$-hull of a frame, establishing an equivalence between categories of proximity morphisms on MT-algebras and frames, with implications for $T_D$-duality.

Contribution

It introduces proximity morphisms between MT-algebras and demonstrates their categorical equivalence to frames via the Funayama envelope viewed as the $T_D$-hull.

Findings

Category of proximity morphisms is equivalent to frames

Generalization of the $T_D$-duality of Banaschewski and Pultr

Spatial ramifications of the equivalence

Abstract

We introduce proximity morphisms between MT-algebras and show that the resulting category is equivalent to the category of frames. This is done by utilizing the Funayama envelope of a frame, which is viewed as the $T_D$-hull. Our results have some spatial ramifications, including a generalization of the $T_D$-duality of Banaschewski and Pultr.