A triple construction on $d$-algebras

TL;DR

This work develops a triple construction on a $d$-algebra $(A;\ast,0)$ by forming the normalizer $A^{\nabla}$ of triples and equipping it with a binary operation $\star$ and the projection $\epsilon(0)$. It proves that, for a $d$-transitive $d$-algebra, the induced order $<$ on $A^{\nabla}$ is a poset and, in suitable cases, yields a $BCK$-algebra, linking generalized $d$-algebras to classical logical algebras. The approach clarifies when $A^{\nabla}$ becomes an algebra under $\star$, how $\epsilon(0)$ acts as a minimal element, and how edge and $d$-transitive properties influence the resulting $BCK$-structure, including examples where a non-$BCK$ algebra induces a $BCK$-algebra on its normalizer. Overall, the paper strategies provide a bridge between $d$-algebra generalizations and $BCK$-type logics via the triple construction.

Abstract

In this note, we consider a triple construction $(\ad;\star,ε(0))$ on a $d$-algebra $(A;\ast,0)$ and investigate some of their properties. Applying this construction to a $d$-transitive $d$-algebra, we show that $(\ad; <)$ is a poset, which induces a $BCK$-algebra.

Theorems & Definitions (21)

• Definition 2.2
• Lemma 2.3
• Proposition 2.4
• Definition 2.5
• Theorem 2.6
• Proposition 3.2
• Theorem 3.3
• proof
• Proposition 3.4
• proof