The Nearly Boolean Nature of Core Regular Double Sone Algebras, CRDSA (Ternary Set Partitions, CRDSA, Embeddings and Dual Equivalences)

TL;DR

This paper explores the nearly Boolean structure of Core Regular Double Stone Algebras (CRDSA) by establishing their lattice, duality with bi-topological spaces, and their subdirect product representations, highlighting their close relation to Boolean algebras.

Contribution

It introduces a duality between CRDSA and bi-topological spaces, and characterizes CRDSA as nearly Boolean through lattice and homomorphism properties.

Findings

CRDSA is isomorphic to C3^J and is a subdirect product of C3.

The category of CRDSA is dually equivalent to core regular double pairwise Stone spaces.

Conditions for bi-continuous maps to have CRDSA homomorphism inverses are established.

Abstract

In "Centre of Core Regular Double Stone Algebra" (CRDSA), many useful results are shown that begin to indicate the nearly Boolean nature of CRDSA which we focus on here. We define the node set lattice through the well known binary operations of ternary set partitions and show the resultant lattice is isomorphic to C3^J where C3 is the 3 element chain CRDSA. We derive that every CRDSA is a subdirect product of C3 similarly as for Boolean algebras and C2. We use these results to show that every Boolean algebra is the center of some CRDSA. Next we show that C3 is primal implying that the variety generated by it is dually equivalent to the category of Boolean algebras. In some sense this is a last step towards our goal of establishing CRDSA as nearly Boolean, but leaves us a bit dissatisfied. Hence we continue by establishing a duality between the category of CRDSA and specifically crafted bi-topological spaces. Towards this end we first establish necessary and sufficient conditions on a pairwise zero-dimensional space such that it will have a CRDSA base B1. We note that these conditions are indicative of how nearly Boolean CRDSA are. For example, if u in B1 is not clopen/complemented then Cl(u) is in B1 and is clopen/complemented. Then we establish necessary and sufficient conditions for a bi-continuous map to have an inverse that is a CRDSA homomorphism again indicating how nearly Boolean CRDSA are, these inverses must respect the appropriate conditions on the boundary of non-clopen elements of B1. In culmination we show the category of core regular double Stone algebras is dually equivalent to the category of what we call core regular double pairwise Stone spaces. We note that the conditions for this duality can easily be relaxed to yield a duality for a less rigid class lattices than CRDSA, bounded distributive pseudo-complemented lattices for example.