# Cancellative set in a semiring that is not multiplicatively closed

1. Sep 2, 2010

Definition: A semigroup is a pair (R,op) where R is a set an op is a binary operation that is closed and associative. A commutative semigroup is a semigroup where op satisfies for all a,b in R, op(a,b) = op(b,a). A monoid is a semigroup where with an identity,e, for op, satisfying for all r in R, op(r,e)=op(e,r)=r. A commutative monoid is a semigroup satisfying the monoid and commutative semigroup laws.

Definition:
A semiring (rig) is a triple (R,+,.) where (R,+) is a commutative monoid, and (R,.) is a semigroup, and where distributivity holds;i.e.
a(b+c) = ab+ac and (b+c)a = ba+ca.

Definition:
Let R=(R,+,.) be a semiring. A cancellable element r satisfies
r+a = r+b implies a = b.
The set of all cancellable elements in R is denoted Can(R).

It is an easy exercise to show that Can(R) is a submonoid of (R,+). However, it does not seem to be multiplicatively closed. If anyone knows or can sketch a proof that would be great! Are there any canonical examples of Rigs, where the cancellative elements are not multiplicatively closed.

2. Sep 6, 2010

### Eynstone

As there may not be any connection between + and . of R, Can(R) need not be closed.

3. Sep 6, 2010