On seminormal semigroups

摘要

Each semigroup, written additively, is a non-zero, commutative, torsionfree cancellative semigroup with 0. Let S be a semigroup with the quotient group G. T is is said to be integral over S if nt∈S for some positive integer n. The set S' of elements t∈G that are integral over S is called the integral closure of S. We say that S is a seminormal semigroup if 2a, 3a∈S or -a∈S for each a∈G. An ideal of S is a nonempty subset I of S such that Is+I for each s∈S. An ideal I of S is prime if I=/S and if x+y∈I implies x∈I or y∈I for x, y∈S. We say that dim(S)=1 if S has one and only one prime ideal. In this paper, we shall consider the properties of seminormal semigroups. In particular, we shall prove that, for a semigroup S with dim S=1, (1) if S is seminormal, then each integral oversemigroup of S is seminormal. (2) Each oversemigroup of S is seminormal if and only if both S is seminormal and S' is a valuation semigroup.