Semigroup Forum Vol. 54 (1997) 75-82 © 1997 Springer-Verlag New York Inc. RESEARCH ARTICLE On Ordered Filters of Implicative Semigroups Y. B. Jun*, J. Meng** and X. L. Xin** Communicated by L. N Shevrin Abstract In this paper, we first study how to generate an ordered filter by a set. Secondly, we discuss prime ordered filters in commutative implicative semigroups, and establish prime and irreducible decompositions The notions of implicative semigroup and ordered filter were introduced by M. W. Chan and K. P. Shum [3]. The first is a generalization of implicative semilattice (see W. C. Nemitz [5] and T. S. Blyth [2])and has a close relation with implication in mathematical logic and set theoretic difference (see G. Birkhoff [1] and H. B. Curry [4]). For the general development of implicative semilattice theory the ordered filters play an important role which is shown by W. C. Nernitz [5]. To be motivated by this, M. W. Chan and K. P. Shum [3] established some elementary properties, and constructed quotient structure of implicative semigroups via ordered filters. To deeply study of implicative semigroups it is undoubtedly necessary to establish more complete theory of ordered filters for it. The aim of the present paper is to discuss ordered filters of commutative implicative semigroups. We study how to generate an ordered filter by a set. Following the idea of general lattice theory, we introduce the notions of prime ordered filters and irreducible ones, and prove that they coincide. Finally we simply discuss prime and irreducible decompositions of ordered filters. We recall some definitions and results. By a negatively partially ordered semigroup (briefly, n.p.o, semigroup), we mean a set 5: with a partial ordering "<" and a binary operation "." such that for all x,y,z E 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 andx.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 ifz-x<y. The operation * is called implication. From now otl, 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 forall x,yE S, that is, (S,-) is a commutative semigroup. In any implicative semigroup (S; <,., *), x * x = y * y and this element is the greatest in S; it will be denoted by 1. • Supported by the Basic Science Research Institute Program, Ministry of Education, 1994, Project No. BSRI-94-1406. • * The authors are heartily thankful to the referee for his valuable comments.