The completeness and congruences of quasi-Boolean algebras

TL;DR

This paper explores the structure, completeness, and congruence properties of quasi-Boolean algebras, a generalization of Boolean algebras relevant to quantum computation logic, providing new characterizations and extension properties.

Contribution

It offers a detailed analysis of finite quasi-Boolean algebras, proves their standard completeness, and characterizes congruence extension from subalgebras to the entire algebra.

Findings

Characterization of finite irreducible quasi-Boolean algebras

Proof of standard completeness of quasi-Boolean algebras

Congruence extension property for quasi-Boolean algebras

Abstract

Quasi-Boolean algebras were introduced as the generalization of Boolean algebras in the setting of quantum computation logic. In this paper, we investigate the completeness and congruences of quasi-Boolean algebras. First, we discuss the number of finite quasi-Boolean algebras and characterize the finite irreducible quasi-Boolean algebras. Second, we show the standard completeness of quasi-Boolean algebras. Finally, we prove that the variety of quasi-Boolean algebras satisfies the congruence extension property and provide a complete characterization of how congruences on a Boolean subalgebra can be extended to the whole quasi-Boolean algebra.