I recently came across an algorithm that works on values assuming that they are draw from a monoid equipped with a total ordering relation. I was wondering if there is a term for such a structure, since it seems related to concepts like Euclidean domains and fields (though the requirements are much less strict). Does this entity have a name? Or is it just "a monoid over totally ordered elements?"
Thanks!
If the underlying order is assumed to be a total order, the terms "totally ordered monoid" or "totally ordered semigroup" seem appropriate. If the underlying order is a partial order, then as was mentioned in @Bill Dubuque's post, the term for this is an "ordered monoid" (or "ordered semigroup" in the case of semigroups).