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A binary operation on a set is a function from the Cartesian product to . A magma (or binary algebraic structure) is a set equipped with a binary operation on it.
A magma is called
unital if it has a neutral element , hence an element in that 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: ;
associative if the binary operation satisfies the associativity condition .
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 , a magma object or magma in is an object of equipped with a multimorphism in . Here the multimorphism from and to is a binary operation in . In particular, for a monoidal category, a magma structure on is a morphism ; and in a closed category, a magma structure on is a morphisms .
The Eckmann-Hilton argument holds for unital magma structures: two compatible ones on a set must be equal 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
Revised on April 21, 2017 03:30:29
by Urs Schreiber