Very true pseudo-BCK algebras

TL;DR

This paper introduces and studies very true operators on pseudo-BCK algebras, exploring their properties, related algebraic structures, and applications to various classes including Smarandache pseudo-BCK algebras.

Contribution

It defines very true operators, investigates their properties, and extends the concept to various algebraic structures and classes, including quotient and Smarandache pseudo-BCK algebras.

Findings

Composition of very true operators commutes iff they are compatible.

Defined pseudo-BCK_{vt,st} algebra with truth-depressing hedges.

Analyzed properties on classes like pseudo-BCK(pP), FL_w, and pseudo-MTL.

Abstract

In this paper we introduce the very true operators on pseudo-BCK algebras and we study their properties. We prove that the composition of two very true operators is a very true operator if and only if they commute. Moreover, given a very true bounded pseudo-BCK algebra $(A,v)$, we define the pseudo-BCK$_{vt,st}$ algebra by adding two truth-depressing hedges operators associated with $v$. We also define the very true deductive systems and the very true homomorphisms and we investigate their properties. Also, given a normal $v$-deductive system of a very true pseudo-BCK algebra $(A,v)$ we construct a very true operator on the quotient pseudo-BCK algebra $A/H$. We investigate the very true operators on some classes of pseudo-BCK algebras such as pseudo-BCK(pP) algebras, FL$_w$-algebras, pseudo-MTL algebras. Finally, we define the $Q$-Smarandache pseudo-BCK algebras and we introduce the notion of a very true operator on $Q$-Smarandache pseudo-BCK algebras.