Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?"


share|cite|improve this question
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).

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.