IJMMS 27:2 (2001) 77–82 PII. S0161171201005907 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON IDEALS OF IMPLICATIVE SEMIGROUPS YOUNG BAE JUN and KYUNG HO KIM (Received 2 October 2000) Abstract. We introduce the notion of ideals in implicative semigroups, and then state the characterizations of the ideals. 2000 Mathematics Subject Classification. 20M12, 06F05, 06A06, 06A12. 1. Introduction. The notions of implicative semigroup and ordered filter were in- troduced by Chan and Shum [3]. The first is a generalization of implicative semilattice (see Nemitz [6] and Blyth [2]) and has a close relation with implication in mathematical logic and set theoretic difference (see Birkhoff [1] and Curry [4]). For the general development of implicative semilattice theory the ordered filters play an important role which is shown by Nemitz [6]. Motivated by this, Chan and Shum [3] established some elementary properties, and constructed quotient structure of implicative semi- groups via ordered filters. Jun et al. [5] discussed ordered filters of implicative semi- groups. In this paper, we introduce the notion of ideals in implicative semigroups. By introducing special subsets of an implicative semigroups, we provide a condition for the special subset to be an ideal. We establish two characterizations of ideals. 2. Preliminaries. We recall some definitions and results. By a negatively partially ordered semigroup (briefly, n.p.o. semigroup) we mean a set S with a partial ordering ≤ and a binary operation · such that for all x,y,z ∈ S , we have (1) (x · y) · z = x · (y · z), (2) x ≤ y implies x · z ≤ y · z and z · x ≤ z · y , (3) x · y ≤ x and x · y ≤ y . An n.p.o. semigroup (S ; ≤, ·) is said to be implicative if there is an additional binary operation ∗ : S × S → S such that for any elements x,y,z of S , (4) z ≤ x ∗ y if and only if z · x ≤ y. The operation ∗ is called implication. From now on, an implicative n.p.o. semigroup is simply called an implicative semigroup. An implicative semigroup (S ; ≤, ·, ∗) is said to be commutative if it satisfies (5) x · y = y · x for all x,y ∈ S , that is, (S, ·) is a commutative semigroup. In any implicative semigroup (S ; ≤, ·, ∗), x ∗ x = y ∗ y for every x,y ∈ S and this element is the greatest element, written 1, of (S, ≤). Proposition 2.1 (see [3, Theorem 1.4]). Let S be an implicative semigroup. Then for every x,y,z ∈ S , the following hold: (6) x ≤ 1,x ∗ x = 1,x = 1 ∗ x, (7) x ≤ y ∗ (x · y),