# Binary operations

## Definitions

A binary operation on a set $S$ is a function from $S \times S$ to $S$. A magma (binary algebraic structure) is a set equipped with a binary operation on it. A magma is unital if it has a neutral element $1$. Some authors mean by ‘magma’ what we call a unital magma (cf. Borceux-Bourn Def. 1.2.1). The Eckmann-Hilton argument holds for unital magma structures: two compatible ones on a set must be equal and commutative.

The term ‘magma’ is from Bourbaki and intends to suggest the fluidity of the concept; special cases include semigroups, quasigroups, groups, and so on. The term ‘groupoid’ is also used, but here that word means something else (see also related discussion at historical notes on quasigroups).

More generally, in any multicategory $M$, a magma object or magma in $M$ is an object $X$ of $M$ equipped with a multimorphism $m: X, X \to X$ in $M$. Here the multimorphism from $X$ and $X$ to $X$ is a binary operation in $M$. In particular, for $M$ a monoidal category, a magma structure on $X$ is a morphism $m\colon X \otimes X \to X$; and in a closed category, a magma structure on $X$ is a morphisms $m\colon X \to [X, X]$.

## Literature

The wikipedia entry magma is quite useful, having a list and a table of various subclasses of magmas, hence of binary algebraic structures.

• eom: magma
• R.H. Bruck, A survey of binary systems, Springer-Verlag 1958

category: algebra

Revised on November 2, 2013 03:29:48 by Zoran Škoda (77.237.114.65)