# Is there a more common term for “hemigroup”?

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?

• Typo in (ii)? [And a small typo in (iv)]. – M. Vinay Apr 11 at 1:17
• Unless my eyes are plying tricks on me, The laws are stated consistently with the book. (i)associative; (ii)commutative; (iii) cancellation; (iv) neutral element. – Steven Thomas Hatton Apr 11 at 1:23
• Okay, so commutativity is: $(x*y) = (y*x)$ (you've written $(y*z)$ instead of $(y*x)$, unless it's my eyes that are playing tricks on me!) : D – M. Vinay Apr 11 at 1:24
• I don't know if this helps much, but it appears to be a commutative monoid with identity. – Gary Moon Apr 11 at 2:02
• @GaryMoon monoid means "semigroup with identity", so "monoid with identity" means "semigroup with identity with identity". – YCor Apr 11 at 9:21

## 1 Answer

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).