Commutative Semigroups Obtained by an Abelian Group and a Generalized•¬-Function

Throughout Z+ denotes the semigroup of all positive integers, Z+ =Z+ U|0) the semigroup of all non-negative integers. The operation in each is the usual addition. Let G be an abelian group and I: G x G-*Z+ be a function satisfying the following conditions: (1) / (a,ft)= / (fta) for all a, /3eG. (2) / («,j8)+J («j8,r)=J («,ftO+ / (ftr) for all a, ft r^G. Let S={(a,*):aEG,iGZt°|. Define an operation on S by («,*) (Atf>=(«&*+>+/(«,£». Then S becomes a semigroup. The S obtained in this manner is denoted by S=(G: /). S is called an 9i-semigroup. A function /: GxG-^Zl satisfying (1), (2) is called an ^"-function on G. As well known, if the above y'-function / satisfies additionally (3) / (s,a)=l (e being the identity of G) for all a<=G (4) For each «gG there is an m<^Z+ such that I(a,ara)^> 0, then / becomes an f-iunction on G and S=(G: 7) does an 9c-semigroup (i. e., a com-mutative archimedean cancellative idempotent-free semigroup [1, 2, 4]). In this paper, we study Tc-semigroups from the standpoint of their archimedeaness. 1. The classification of-ft-semigroups. Throughout S={G: 1} denotes the 5J-seniigroup obtained by an abelian group G and ^-function / on G, e does the identity element of G. First we mention the following: Proposition 1. An yt-semigroup S = (G: /) is a commutative cancellative semigroup. Proof. Is is straightforward. Lemma 2. I (a,e)=I (£,«)= / (s,s) for all ae.G. Proof. Put j3=j-=e in the condition (2). The following Lemma 3 and Proposition 4 show that the having idempotents, archi-medeaness of 9J-semigroups. Lemma 3. Let S=(G: I) be an ^-semigroup. Under this assumption the following are equivalent