On some generalizations of BCC-algebras

TL;DR

This paper introduces and characterizes weak BCC-algebras, exploring their properties when a specific associativity condition holds within branches, and examines relationships among different implicative variants.

Contribution

It defines weak BCC-algebras with branch-specific associativity and characterizes branchwise commutative and implicative cases, expanding the understanding of BCC-algebra generalizations.

Findings

Weak BCC-algebras satisfy (xy)z=(xz)y only within the same branch.

Characterization of branchwise commutative weak BCC-algebras.

Connections between various implicative weak BCC-algebra types.

Abstract

We describe weak BCC-algebras (also called BZ-algebras) in which the condition $(xy)z=(xz)y$ is satisfied only in the case when elements $x,y$ belong to the same branch. We also characterize branchwise commutative and branchwise implicative weak BCC-algebras satisfying this condition. We also describe connections between various types of implicative weak BCC-algebras.