nLab magma

Binary operations

Context

Algebra

[[!include higher algebra - contents]]

Binary operations

Definitions

A binary operation on a set $S$ is a function $(-)\cdot (-) \colon S \times S \to S$ from the Cartesian product $S \times S$ to $S$. A magma (or binary algebraic structure, or, alternatively, a mono-binary algebra) $(S,\cdot)$ is a set equipped with a binary operation on it.

A magma is called

• unital if it has a neutral element $1$, hence an element $1 \in S$ in that $1 \cdot x = x = x \cdot 1$ Some authors mean by ‘magma’ what we call a unital magma (cf. Borceux-Bourn Def. 1.2.1).

• commutative if the binary operation takes the same value when its two arguments are interchanged: $x \cdot y = y \cdot x$;

• associative if the binary operation satisfies the associativity condition $(x \cdot y) \cdot z = x \cdot (y \cdot z)$.

The term ‘magma’ is from Bourbaki and intends to suggest the fluidity of the concept; special cases include semigroups/monoids, 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 $m\colon X \to [X, X]$.

Properties

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

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

Last revised on December 22, 2017 at 17:49:43. See the history of this page for a list of all contributions to it.