Noetherian and Artinian ordered groupoids—semigroups

Chain conditions, finiteness conditions, growth conditions, and other forms of finiteness, Noetherian rings and Artinian rings have been systematically studied for commutative rings and algebras since 1959. In pursuit of the deeper results of ideal theory in ordered groupoids (semigroups), it is necessary to study special classes of ordered groupoids (semigroups). Noetherian ordered groupoids (semigroups) which are about to be introduced are particularly versatile. These satisfy a certain finiteness condition, namely, that every ideal of the ordered groupoid (semigroup) is finitely generated. Our purpose is to introduce the concepts of Noetherian and Artinian ordered groupoids. An ordered groupoid is said to be Noetherian if every ideal of it is finitely generated. In this paper, we prove that an equivalent formulation of the Noetherian requirement is that the ideals of the ordered groupoid satisfy the so-called ascending chain condition. From this idea, we are led in a natural way to consider a number of results relevant to ordered groupoids with descending chain condition for ideals. We moreover prove that an ordered groupoid is Noetherian if and only if it satisfies the maximum condition for ideals and it is Artinian if and only if it satisfies the minimum condition for ideals. In addition, we prove that there is a homomorphism π of an ordered groupoid (semigroup) S having an ideal I onto the Rees quotient ordered groupoid (semigroup) S / I . As a consequence, if S is an ordered groupoid and I an ideal of S such that both I and the quotient groupoid S / I are Noetherian (Artinian), then so is S . Finally, we give conditions under which the proper prime ideals of commutative Artinian ordered semigroups are maximal ideals.