Boolean perspectives of idioms and the Boyle derivative

TL;DR

This paper explores boolean properties of idioms using the Boyle-derivative, providing conditions for nuclei to form complete boolean algebras and analyzing complemented idioms.

Contribution

It introduces an idiomatic analysis of boolean properties of idioms via the Boyle-derivative, connecting lattice theory with module theory concepts.

Findings

Conditions for nuclei to form complete boolean algebras

Characterization of complemented idioms

Analysis of the Boyle-derivative's role in boolean properties

Abstract

We are concerned with the boolean or more general with the complemented properties of idioms (complete upper-continuous modular lattices). In [Simmons&Cantor] the author introduces a device which captures in some informal speaking how far the idiom is from be complemented, this device is the Cantor-Bendixson derivative. There exists another device that captures some boolean properties, the so-called Boyle-derivative, this derivative is an operator on the assembly (the frame of nuclei) of the idiom. The Boyle-derivative has its origins in module theory. In this investigation we produce an idiomatic analysis of the boolean properties of any idiom using the Boyle-derivative, we give conditions on a nucleus $j$ such that $[j, tp]$ is a complete boolean algebra. We also explore some properties of nuclei $j$ such that $A_{j}$ is a complemented idiom.