States on Pseudo-bck Algebras

The notion of a state is an analogue of a probability measure and was first introduced by Kôpka and Chovanec for MV-algebras and by Riec̆an for BLalgebras. The states have also been studied for different types of non-commutative fuzzy structures such as pseudo-MV algebras, pseudo-BL algebras, bounded R`monoids, residuated lattices and pseudo-BCK semilattices. In this paper we investigate the states on pseudo-BCK algebras and show that Georgescu’s original problem in [10] for pseudo-BL algebras has a negative solution for good pseudoBCK algebras. We prove that every Bosbach state on a pseudo-BCK algebra is a Riec̆an state and that every Riec̆an state on a good pseudo-BCK algebra with pseudo-double negation is a Bosbach state. In contrast to the case of pseudo-BL algebras, we show that there exist linearly ordered pseudo-BCK algebras having no Bosbach states.