# Is there a formal term for the algebraic structure equivalent to Thurston's hemigroup without identity? [duplicate]

This question is not a duplicated of https://math.stackexchange.com/a/3183330/342834 . It is a follow-on, asking about the distinction between the subject of that post, and the subject of this post.

In The Number System by Thurston the author introduces an algebraic structure he calls a hemigroup. The laws of a hemigroup are:

• (i) $$\left(x\odot y\right)\odot z=x\odot \left(y\odot z\right),$$
• (ii) $$\left(x\odot y\right)=\left(y\odot x\right),$$
• (iii) $$\left(x\odot y\right)=\left(x\odot z\right)\implies{y=z},$$
• (iv) $$\exists_e e\odot e=e.$$

I have been told this defines a cancellative commutative monoid. I will retain the uncommon designation hemigroup for now.

Thurston then introduces entities he calls dyads which are sets of ordered pairs formed of elements of the hemigroup constituting equivalence classes of the ordered pairs. I write this as

$$\left[\![a,b\right]\!]\equiv\{ \left\backepsilon{x\odot b=y\odot a}\}.$$

The algebra of these dyads satisfies the laws of a commtative group. One such commutative group is what BBFSK call the module of integers. In that source, however, this module is constructed of residue classes of ordered pairs from an algebraic structure which amounts to Thurston's hemigroup without the existence of an identity element. That is

• (i) $$\left(x\odot y\right)\odot z=x\odot \left(y\odot z\right),$$
• (ii) $$\left(x\odot y\right)=\left(y\odot x\right),$$
• (iii) $$\left(x\odot y\right)=\left(x\odot z\right)\implies{y=z}.$$

But those authors do not formally nominate this structure. It is the additive algebraic structure of the natural numbers, which by traditional American definition are $$\mathbb{N}\equiv\{1,2,3,\dots\}$$. Thurston's hemigroup is the additive structure of the whole numbers, which by traditional American definition are $$\mathbb{N}_0\equiv \mathbb{N}\cup\{0\}.$$

Both developments produce the same integral domain, but from different underlying algebraic structures.

I am asking if the algebraic structure consisting of a hemigroup sans identity has a common mathematical name. If so, what is that name?

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• Thank you for that clarification regarding the idempotent element. Thurston, in fact, never uses that element directly, but only in defining the isomorphism between dyads of the form $\left[\![x,e\right]\!]$ and the underlying hemigroup, as well as defining the identity dyad $\varepsilon\equiv \left[\![e,e\right]\!]$. For the latter $e$ is certainly unnecessary. – Steven Thomas Hatton Apr 24 at 16:15