nLab
magma

Binary operations

Definitions

A binary operation on a set SS is a function from S×SS \times S to SS. 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 11. 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 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].

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)