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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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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
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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

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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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