On seminormal semigroups

Abstract.

Each semigroup, written additively, is a non-zero, commutative, torsion-free cancellative semigroup with 0. Let S be a semigroup with the quotient group G. T is called an oversemigroup of S if T is a subsemigroup of G containing S. An element $t\in T$ is said to be integral over S if $nt \in S$ for some positive integer n. The set S' of elements $t \in G$ that are integral over S is called the integral closure of S. We say that S is a seminormal semigroup if $2\alpha, 3\alpha \in S$ for $\alpha \in G$, we have $\alpha \in S$. Also, S is called a valuation semigroup if either $\alpha \in S$ or $-\alpha \in S$ for each $\alpha \in G$. An ideal of S is a non-empty subset I of S such that $I \supset s + I$ for each $s\in S$. An ideal I of S is prime if $I \ne S$ and if $x + y \in I$ implies $x \in I$ or $y \in I$ for $x,y \in S$. 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.