d-Boolean algebras and their bitopological representation
Hang Yang, Dexue Zhang
TL;DR
This work extends Stone duality to the bitopological setting by introducing $d$-Boolean algebras as algebraic counterparts to bitopological spaces and by formulating a duality with $d$-Boolean spaces. It builds a web of adjunctions and equivalences among distributive lattices, $d$-lattices, $d$-frames, and their ideals, culminating in a duality ${f dBA}^{op}\\simeq\ {f dBoolSp}$ via the spectrum construction $ ext{dSpec}$. The paper clarifies when $d$-frames are spatial and connects to Jakl and Jung–Moshier results on coherent and $d$-zero-dimensional frames, providing a point-free/point-set bridge for bitopology. This duality broadens algebraic representations for bitopological structures and has potential implications for logic and topology in bitopological contexts.
Abstract
We present a Stone duality for bitopological spaces in analogy to the duality between Stone spaces and Boolean algebras, in the same vein as the duality between d-sober bitopological spaces and spatial d-frames established by Jung and Moshier. Precisely, we introduce the notion of d-Boolean algebras and prove that the category of such algebras is dually equivalent to the category of compact and zero-dimensional bitopological spaces satisfying the T0 separation axiom.