Binary operations

Binary operations


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

A magma is called

  • unital if it has a neutral element 11, hence an element 1S1 \in S in that 1x=x=x11 \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: xy=yxx \cdot y = y \cdot x;

  • associative if the binary operation satisfies the associativity condition (xy)z=x(yz)(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 MM, a magma object or magma in MM is an object XX of MM equipped with a multimorphism m:X,XXm: X, X \to X in MM. Here the multimorphism from XX and XX to XX is a binary operation in MM. In particular, for MM a monoidal category, a magma structure on XX is a morphism m:XXXm\colon X \otimes X \to X; and in a closed category, a magma structure on XX is a morphisms m:X[X,X]m\colon X \to [X, X].


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


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.