Some properties of pseudo-BCK- and pseudo-BCI-algebras

TL;DR

This paper explores properties of pseudo-BCK- and pseudo-BCI-algebras, showing their relation to residuated po-monoids, characterizing certain decompositions, and establishing their algebraic congruence properties.

Contribution

It demonstrates that pseudo-BCI-algebras are residuation subreducts of semi-integral residuated po-monoids and characterizes their structure as direct products, also proving their congruence modularity.

Findings

Pseudo-BCI-algebras are residuation subreducts of semi-integral residuated po-monoids.

Certain pseudo-BCI-algebras decompose into direct products of pseudo-BCK-algebras and groups.

The quasivariety of pseudo-BCI-algebras is relatively congruence modular.

Abstract

Pseudo-BCI-algebras generalize both BCI-algebras and pseudo-BCK-algebras, which are a non-commutative generalization of BCK-algebras. In this paper, following [J.G. Raftery and C.J. van Alten, Residuation in commutative ordered monoids with minimal zero, Rep. Math. Log. 34 (2000) 23-57], we show that pseudo-BCI-algebras are the residuation subreducts of semi-integral residuated po-monoids and characterize those pseudo-BCI-algebras which are direct products of pseudo-BCK-algebras and groups (regarded as pseudo-BCI-algebras). We also show that the quasivariety of pseudo-BCI-algebras is relatively congruence modular; in fact, we prove that this holds true for all relatively point regular quasivarieties which are relatively ideal determined, in the sense that the kernels of relative congruences can be described by means of ideal terms.