Involutive filters of pseudo-hoops

Abstract

In this this paper we introduce the notion of involutive filters of pseudo-hoops, and we emphasize their role in the probability theory on these structures. A characterization of involutive pseudo-hoops is given and their properties are investigated. We give characterizations of involutive filters of a bounded pseudo-hoop and we prove that in the case of bounded Wajsberg pseudo-hoops the notions of fantastic and involutive filters coincide. One of main results consists of proving that a normal filter $F$ of a bounded pseudo-hoop $A$ is involutive if and only if $A/F$ is an involutive pseudo-hoop. It is also proved that any Boolean filter of a bounded Wajsberg pseudo-hoop is involutive. The notions of state operators and state-morphism operators on pseudo-hoops are introduced and the relationship between these operators are investigated. For a bounded Wajsberg pseudo-hoop we prove that the kernel of any state operator is an involutive filter.