Fei Xie and Hongxing Liu
In this paper, we introduce the notion of Ehoops, which are generalizations of hoops. Unlike the hoop, an Ehoop does not necessarily have a top element. The notions of ideals and filters in Ehoops are defined. In Ehoops, both ideals and filters can be used to define the congruences. It is proved that if an Ehoop A satisfies the double negation property, there is a one-to-one correspondence between the set of all ideals of A and the set of all congruences on A. Prime ideal theorem on Ehoops is also given. In addition, we define the notions of implicative filters and positive implicative filters of Ehoops and investigate the quotient algebras induced by (positive) implicative filters.
Keywords: Ehoop, hoop, idempotent element, congruence, ideal, filter