A characterization of Artinian $l$-semigroups
Kentaro Murata · Proceedings of the Japan Academy Series A Mathematical Sciences · 1971
The aim of the present note is to generalize Artin's well-known equivalence relation (quasi-equal relation) introduced in commutative rings or some sorts of commutative /-semigroups, ) and to give a characterization o such/-semigroups by a system of valuations defined on some quotient/-semigroups with compactly generated cones.1o Let S be a conditionally complete and commutative /-semi- group with unity quantity e, and let I be the cone (integral part) of S. We suppose throughout this paper that I is compactly generated by a compact generator system 27 containing e (cf.[7]), and that S is a quotient semi-group o I by 27, that is, every element x o 27 is inverti- ble in.S and every element c of S can be written as c=ax-, where a e I and x e 27.If a compactly generated/-semigroup I with a compact generator system 27 is given, we can prove that there exists a quotient /-semigroup of I by 27, if and only if the following two conditions hold or I and 27: (i) for any two elements x and y of 27, there exists an ele- ment a of I such that axy is in 27, and (ii) every element of 2 satisfies the cancellation law.The lattice-structure is naturally introduced in the quotient semigroup, and such a quotient /-semigroup is uniquely