Abstract
We discuss totally ordered monoids (or tomonoids, for short) that are commutative, positive, and finitely generated. Tomonoids of this kind correspond to certain preorders on free commutative monoids. In analogy to positive cones of totally ordered groups, we introduce direction cones to describe the preorders in question and we establish between both notions a Galois connection. In particular, we show that any finitely generated positive commutative tomonoid is the quotient of a tomonoid arising from a direction cone. We furthermore have a closer look at formally integral tomonoids and at nilpotent tomonoids. In the latter case, we modify our approach in order to obtain a description that is based on purely finitary means.
Highlights
An interest in totally ordered monoids, or tomonoids as we say in accordance with [4], is present in diverse fields
For a given n 1, let P be the set of all monomial preorders on natural order on (Nn) and let C be the set of all direction cones in Zn
Each finitely generated commutative monoid L is a quotient of Nn
Summary
An interest in totally ordered monoids, or tomonoids as we say in accordance with [4], is present in diverse fields. A compatible, positive total order needs to be taken into account To this end, we do not directly consider the congruences in question, but instead the preorders induced by the total order of tomonoids on their associated free monoids. We establish a Galois connection for this more special situation and we show that any nilpotent finite, positive, commutative tomonoid is the quotient of a tomonoid induced by a direction f-cone.
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