![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
etbI have an operation that is associative, commutative, and has an identity "1", but does not have inverses. I also need division, e.g., abcdf / bf = acd, but as a partial operation, because foo / bar is only defined when every 'bar' element appears in 'foo'. Thus 1 / a is not defined (if it were I'd have inverses, and thus a group). Is there a better description than "commutative monoid [with FREE! BONUS! partial division]"?
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2009-03-10 12:09 am (UTC)ab = ac implies b = c
as otherwise the division is well defined. This is called having the "cancellation property" according to Wikipedia (http://en.wikipedia.org/wiki/Monoid).
One example of something satisfying your conditions is a sub-monoid of a commutative group. It's conceivable that those are the only examples, actually, as you could try to build the group from the given monoid in the "obvious" way (i.e. form the Grothendieck group), the tricky bit being showing the resulting map is injective...hmm, Wikipedia claims this is indeed the case (http://en.wikipedia.org/wiki/Monoid#Properties).
no subject
Date: 2009-03-10 12:11 am (UTC)