In The Number System by Thurston the author introduces an algebraic structure he calls a "hemigroup". It doesn't appear to be a very common usage. The laws of a hemigroup are:
- (i) $\left(x*y\right)*z=x*\left(y*z\right)$
- (ii) $\left(x*y\right)=\left(y*x\right)$
- (iii) $\left(x*y\right)=\left(x*z\right)\implies{y=z}$
- (iv) $\exists_e e*e=e$
Apparently there are other definitions for the same term. Is there a more common term than "hemigroup" for this kind of structure?
This is usually called a cancellative commutative monoid. Note that in the presence of (iii), (iv) is equivalent to saying that $e$ is a (left) identity, so these axioms just say you have a commutative associative operation with an identity element (i.e., a commutative monoid) which satisfies the cancellation axiom (iii).