The completeness and congruences of quasi-Boolean algebras
Xiaohao Liu, Heyan Wang, Wenjuan Chen
TL;DR
Quasi-Boolean algebras ($\mathbb{QB}$) generalize Boolean algebras for quantum logical settings, prompting questions about completeness and congruences. The authors establish a finite-algebra classification and a standard completeness result by reducing to flat and Boolean quotients via $Q/\chi$ and $Q/\tau$ and by identifying generating algebras ($\mathbf{F_3}$, $\mathbf{4}$, $\bar{\mathbf{4}}$). They prove that the CEP holds for $\mathbb{QB}$ by combining CEPs for $\mathbb{B}$ and $\mathbb{FQB}$ and by giving explicit constructions of extended congruences from data on $\mathscr{R}(Q)$ and $\mathscr{IR}(Q)$. These results provide a structural blueprint for how the Boolean and flat components control the equational theory and congruence lattice of QB-algebras, informing semantic foundations for quantum logic.
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.