Involutive filters of pseudo-hoops

TL;DR

This paper introduces involutive filters in pseudo-hoops, explores their properties, and links them to probability theory, providing characterizations and relationships with other filter types and state operators.

Contribution

It defines involutive filters of pseudo-hoops, characterizes them, and establishes their connection with probability structures and state operators in these algebraic systems.

Findings

Involutive filters are characterized within pseudo-hoops.

Normal filters are involutive iff the quotient is involutive.

Boolean filters in Wajsberg pseudo-hoops are involutive.

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.