States and internal states on semihoops

Abstract

In this paper, we investigate states and internal states on bounded semihoops. First, we introduce Bosbach states and Riečan states on bounded semihoops. We derive that Bosbach states are Riečan states on bounded semihoops, while the converse is not always true. Furthermore, we show that Bosbach states and Riečan states on semihoops with Glivenko property are essentially the very same thing. In particular, we prove that Riečan states on a bounded semihoop L are reduced to Riečan states on the set $\mathrm{Reg}(L)$ of all regular elements in L, where $\mathrm{Reg}(L)$ forms a semihoop with double negation property. The same holds true for Bosbach states whenever L is a semihoop with Glivenko property. Our results generalize the existing ones found in the literature. Moreover, to treat a variant of the concept of states within the framework of universal algebras, we introduce internal states on bounded semihoops, which preserve the usual properties of states. We characterize two special types of semihoops using internal states. Also, we discuss relations between internal states and states on bounded semihoops. In addition, we introduce and investigate state filters in state semihoops. In particular, we conclude that the set of all prime state filters in state $\sqcup$-semihoops is a compact $T_{0}$ topological space.