Some Types of Filters in Hoops

Abstract

In this paper we consider fundamental properties of some types of filters (implicative, positive implicative and fantastic filters) of hoops and prove that for any hoop $A$ and filter $F$ of $A$,\begin{quote}(a) $F$ is an implicative filter if and only if $A/F$ is a relatively pseudo-complemented semi lattice, that is, Brouwerian semi lattice,(b) $F$ is a positive implicative filter if and only if $A/F$ is a $\{\wedge, \vee, \to, 1\}$-reduct of Heyting algebra,(c) $F$ is a fantastic filter if and only if $A/F$ is a Wajsberg hoop.\end{quote} Moreover we show that, for any filter of a hoop, it is a positive implicative filter if and only if it is an implicative and fantastic filter.