Filters on some classes of quantum B-algebras

TL;DR

This paper explores filters in quantum B-algebras and pseudo-hoops, establishing embeddings and conditions for reducibility, and extends results on residuated structures to noncommutative settings.

Contribution

It introduces new results on filters in quantum B-algebras and pseudo-hoops, including embeddings and reducibility criteria, extending existing theorems to noncommutative residuated structures.

Findings

Embedding of polar products into pseudo-hoops

Conditions for pseudo-hoop subdirect reducibility

Extension of Kondo and Turunen's result to noncommutative residuated structures

Abstract

In this paper, we continue the study of quantum B-algebras with emphasis on filters on integral quantum B-algebras. We then study filters in the setting of pseudo-hoops. First, we establish an embedding of a cartesion product of polars of a pseudo-hoop into itself. Second, we give sufficient conditions for a pseudohoop to be subdirectly reducible. We also extend the result of Kondo and Turunen to the setting of noncommutative residuated $\vee$-semilattices that, if prime filters and $\vee$-prime filters of a residuated $\vee$-semilattice $A$ coincide, then $A$ must be a pseudo MTL-algebra.