Noetherian and Artinian ordered groupoids—semigroups

Chain conditions, finiteness conditions, growth conditions, andother forms of finiteness, Noetherian rings and Artinian ringshave been systematically studied for commutative ringsand algebras since 1959. In pursuit of the deeper results of idealtheory in ordered groupoids (semigroups), it is necessary to studyspecial classes of ordered groupoids (semigroups). Noetherianordered groupoids (semigroups) which are about to beintroduced are particularly versatile. These satisfy acertain finiteness condition, namely, that every ideal of theordered groupoid (semigroup) is finitely generated. Our purposeis to introduce the concepts of Noetherian and Artinian orderedgroupoids. An ordered groupoid is said to be Noetherian if everyideal of it is finitely generated. In this paper, we prove that anequivalent formulation of the Noetherian requirement is that theideals of the ordered groupoid satisfy the so-called ascendingchain condition. From this idea, we are led in a natural way toconsider a number of results relevant to ordered groupoids withdescending chain condition for ideals. We moreover prove that anordered groupoid is Noetherian if and only if it satisfies themaximum condition for ideals and it is Artinian if and only if itsatisfies the minimum condition for ideals. In addition, we provethat there is a homomorphism π of an ordered groupoid(semigroup) S having an ideal I onto the Reesquotient ordered groupoid (semigroup) S/I. As aconsequence, if S is an ordered groupoid and I an ideal ofS such that both I and the quotient groupoid S/I are Noetherian (Artinian), then so is S. Finally, we giveconditions under which the proper prime ideals of commutativeArtinian ordered semigroups are maximal ideals