A binary operation on a set is a function from the Cartesian product to . A magma (or binary algebraic structure, or, alternatively, a mono-binary algebra) is a set equipped with a binary operation on it.
A magma is called
unital if it has a neutral element; that is, an element such that . Some authors mean by ‘magma’ what we call a unital magma (cf. Borceux-Bourn Def. 1.2.1). One can consider one-sided unital elements separately: and/or . Note that units may be far from unique.
commutative if the binary operation takes the same value when its two arguments are interchanged: .
associative if the binary operation satisfies the associativity condition .
invertible if it has an inverse element.
an absorption magma if it has an element such that the binary operation satisfies the absorption condition: .
a quasigroup if one-sided multiplication by any element is a bijection.
A magma has a square root function if for all , and .
The term ‘magma’ is from Bourbaki and is intended to suggest the fluidity of the concept; special cases include unital magmas, 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 morphism .
Every magma has a morphism called the square and defined as for all .
An action of a set on another set is a function . So this means that a magma is just an action of a set on itself.
There exists a function on the binary operation set called the braiding that takes every binary operation on the set to its opposite binary operation, where for every magma operation and for all elements in , . The set with the opposite binary operation is the opposite magma of . is an involution; the opposite of an opposite magma is the original magma itself; this follows from the fact that Set is a symmetric monoidal category. Fixed points of are called commutative binary operations.
The free magma on one generator is a model of nesting parentheses around a magma, and is important in the study of higher category theory as composition of morphisms in (n,r)-categories and (infinity,n)-categories are not associative, but rather satisfy coherence laws such as the pentagon identity which relate the various ways to nest parentheses around the tensor product magma in an (infinity,n)-category with respect to homotopy equivalence. Many other higher categorical objects have coherence theorems which also deal with nested parentheses around a binary operation.
In foundations of mathematics such as intensional Martin-Loef dependent type theory, where functions do not preserve equality, one could distinguish between ordinary magmas as defined above, and extensional magmas, whose binary operation preserves equality.
For a magma and for all elements , a magma is left extensional if implies , and a magma is right extensional if implies . A magma is extensional if it is both left and right extensional.
On the history of the notion:
The wikipedia entry magma is quite useful, having a list and a table of various subclasses of magmas, hence of binary algebraic structures.
Formalization of magmas as mathematical structures in proof assistants:
in a context of plain Agda:
Last revised on August 21, 2024 at 02:20:21. See the history of this page for a list of all contributions to it.