Some authors take a magma to be unital by default (cf. Borceux-Bourn Def. 1.2.1).

There is also a possibly empty version, where the identity element is replaced with a constant function$1:S \to S$ such that for all $x,y \in S$, $1(x)\cdot y = y$ and $x\cdot 1(y) = x$.

Properties

The Eckmann-Hilton argument holds for unital magmas: two compatible ones on a set must be equal, associative and commutative.