A totally ordered monoid-or tomonoid, for short-is a commutative semigroup with identity S equipped with a total order ≤s that is translation invariant, i.e., that satisfies: x,y,z ∈ S, x ≤ s y => x + z ≤ s y + z. We call a tomonoid that is a quotient of some totally ordered free commutative monoid formally integral. Our most significant results concern characterizations of this condition by means of constructions in the lattice Z~n that are reminiscent of the geometric interpretation of the Buchberger algorithm that occurs in integer programming. In particular, we show that every two-generator tomonoid is formally integral. In addition, we give several (new) examples of tomonoids that are not formally integral, we present results on the structure of nil tomonoids and we show how a valuation-theoretic construction due to Hion reveals relationships between formally integral tomonoids and ordered commutative rings satisfying a condition introduced by Henriksen and Isbell.
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机译：简而言之，一个完全有序的单半体或类同子素是一个交换半群，恒等式S的总阶≤s是平移不变的，即满足：x，y，z∈S，x≤sy => x + z≤sy + z。我们称一个类固醇，是一些完全有序的自由交换单半体形式积分的商。我们最重要的结果涉及通过晶格Z〜n中的结构表征此条件，这种结构让人联想到整数编程中发生的Buchberger算法的几何解释。特别是，我们证明了每个两发电机类同质子在形式上都是整数。此外，我们给出了几个非形式上完整的类同型素实例，我们给出了零类类同型素结构的结果，并展示了因Hion引起的估值理论构造如何揭示形式上类同型类同质素与有序交换环之间的关系。 Henriksen和Isbell提出的条件。
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