On co-annihilators in hoops

Abstract

In this paper, we introduce the notion of co-annihilator in hoops and investigate some related properties of them. Then we prove that the set of filters $\mathcal{F}\left(\mathcal{A}\right)$ form two pseudo-complemented lattices (with ∗ and ⊤) that if A has (DNP), then the two pseudo-complemented lattices are the same. Moreover, by defining the operation → on the lattice $\mathcal{F}\left(\mathcal{A}\right)$ , we prove that $\mathcal{F}\left(\mathcal{A}\right)$ is a Heyting algebra and by defining of the product operation, we show that $\mathcal{F}\left(\mathcal{A}\right)$ is a bounded hoop. Finally, we define the C - Ann (A) to be the set of all co-annihilators of A, then we have that it had made a Boolean algebra. Also, we give an extension of a filter, which leads to a useful characterization of α-filters on hoops. For instance, we obtain a series of characterizations of α-filters. In addition, we show that there are no non-trivial α-filters on hoop-chains. That implies the structure of all α-filters contains only trivial α-filters on hoops. On hoops, we prove that the set of all α-filters is a pseudo-complemented lattice. Moreover, the structure of all α-filters can form a Boolean algebra under certain conditions.