A binary operation on a set is a function from to . 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 . 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 , 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 wikipedia entry magma is quite useful, having a list and a table of various subclasses of magmas, hence of binary algebraic structures.