Involutive filters of pseudo-hoops

In this paper, we introduce the notion of involutive filters of pseudo-hoops, and we emphasize their role in the probability theory on these structures. Characterizations of involutive pseudo-hoops are given, and their properties are investigated. We give a characterization of involutive filters of a good pseudo-hoop, and we prove that in the case of good pseudo-hoops, the sets of fantastic and involutive filters are equal. One of the 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 implicative filter of a bounded pseudo-hoop is involutive and 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 is investigated. For a bounded Wajsberg pseudo-hoop, we prove that the kernel of any state operator is an involutive filter.