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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?"


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It's probably only an interesting structure if the ordering is in some way compatible with the monoid operation. Is it? And if so, how? – Henning Makholm Sep 6 '11 at 1:12
Yes ordered monoid / semigroup, presuming you mean order respecting operations. One should always Google the obvious terms before asking a question, since more focused questions usually yield more helpful answers. – Bill Dubuque Sep 6 '11 at 1:13
@Bill Dubuque- My apologies if this was too obvious. I had indeed looked for this structure, but since I didn't know the right term I didn't find it. Thanks for the tip! – templatetypedef Sep 6 '11 at 3:22
up vote 2 down vote accepted

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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I have to disagree with both this answer and @Bill Dubuque's post. According to wikipedia (and to the literature) an ordered semigroup is a semigroup together with a compatible partial order. – J.-E. Pin Jan 10 at 11:35
@JEPin Hmm, that's a good point. I suppose the term "totally ordered monoid/semigroup" would be more appropriate. – templatetypedef Jan 10 at 17:07
Yes, I fully agree. – J.-E. Pin Jan 10 at 17:51
@J.-E.Pin Thanks for pointing this out. Answer updated! – templatetypedef Jan 10 at 18:29

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