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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?

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    $\begingroup$ Typo in (ii)? [And a small typo in (iv)]. $\endgroup$ – M. Vinay Apr 11 at 1:17
  • $\begingroup$ 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. $\endgroup$ – Steven Thomas Hatton Apr 11 at 1:23
  • $\begingroup$ 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 $\endgroup$ – M. Vinay Apr 11 at 1:24
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    $\begingroup$ I don't know if this helps much, but it appears to be a commutative monoid with identity. $\endgroup$ – Gary Moon Apr 11 at 2:02
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    $\begingroup$ @GaryMoon monoid means "semigroup with identity", so "monoid with identity" means "semigroup with identity with identity". $\endgroup$ – YCor Apr 11 at 9:21
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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).

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