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, (a) F is an implicative filter if and only if A/F is a relatively pseudo-complemented semilattice, that is, Brouwerian semilattice; (b) F is a positive implicative filter if and only if A/F is a {Λ, V, →, 1}reduct of Heyting algebra; (c) F is a fantastic filter if and only if A/F is a Wajsberg hoop. 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.