Unital magma

Remarks

The identity is unique: if $e$ and $e'$ are identities, then $e = e \cdot e' = e'$.

Example: the set $\set{e, a, b}$ with $e \cdot x = x \cdot e = x$ for all $x$ and $u \cdot v = e$ for $u, v \in \set{a, b}$. Both identity equations hold by construction, so this is a unital magma. It is not associative: $(a \cdot a) \cdot b = e \cdot b = b$, but $a \cdot (a \cdot b) = a \cdot e = a$.

Non-example: $(\Z, -)$ has no identity. If $e$ were an identity, then $e - x = x$ for all $x \in \Z$, so $e = 2x$ for every $x$, which is impossible.

In Mathlib the class MulOneClass is a unital magma written multiplicatively.