1
$\begingroup$

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!

  • 0
    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?2011-09-06
  • 1
    Yes [ordered monoid / semigroup,](http://en.wikipedia.org/wiki/Ordered_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.2011-09-06
  • 0
    @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!2011-09-06

1 Answers 1

2

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).

  • 0
    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.2016-01-10
  • 0
    @JEPin Hmm, that's a good point. I suppose the term "totally ordered monoid/semigroup" would be more appropriate.2016-01-10
  • 0
    Yes, I fully agree.2016-01-10
  • 0
    @J.-E.Pin Thanks for pointing this out. Answer updated!2016-01-10