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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!

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    @Bill Dubuque- My apologies if this was too obvious. I had indeed loo$k$ed for this structure, but since I didn't $k$now the right term I didn't find it. Thanks for the tip!2011-09-06

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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).

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    @J.-E.Pin Thanks for pointing this out. Answer updated!2016-01-10