Filters on some classes of quantum B-algebras
Michal Botur, Jan Paseka

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 -semilattices that, if prime filters and -prime filters of a residuated -semilattice coincide, then must be a pseudo MTL-algebra.
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