On state pseudo-hoops

Recently, the bounded pseudo-hoops with internal states have been defined and studied. Some of results are based on the hypothesis that a bounded pseudo-hoop has the Glivenko and (mN) properties. In this paper we show that this hypothesis is superfluous, namely every good pseudo-hoop satisfies the Glivenko and (mN) properties and those results are reformulated without that hypothesis. The pointed pseudo-hoops are introduced and studied and the notion of state operator has been generalized for the case of unbounded pseudo-hoops. We define the Bosbach and Riečan states on pointed pseudo-hoops and we make some considerations regarding the relationship between the state operators and states on these structures.

Local pseudo-BCK algebras with pseudo-product

Pseudo-BCK algebras were introduced by G. Georgescu and A. Iorgulescu as a generalization of BCK algebras in order to give a corresponding structure to pseudo-MV algebras, since the bounded commutative BCK algebras correspond to MV algebras. Properties of pseudo-BCK algebras and their connections with other fuzzy structures were established by A. Iorgulescu and J. Kühr. The aim of this paper is to define and study the local pseudo-BCK algebras with pseudo-product. We will also introduce the notion of perfect pseudo-BCK algebras with pseudo-product and we will study their properties. We define the radical of a bounded pseudo-BCK algebra with pseudo-product and we prove that it is a normal deductive system. Another result consists of proving that every strongly simple pseudo-hoop is a local bounded pseudo-BCK algebra with pseudo-product.