  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.
