Very true pseudo-BCK algebras

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.