Home > Magma (algebra)

The term magma for this kind of structure was introduced by Bourbaki, however, the term groupoid is a very common alternative. Unfortunately, the term groupoid also refers to an entirely different kind of algebraic concept described at Groupoid.

1 Types of magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include

• quasigroups— nonempty magmas where division is always possible;
• loops—quasigroups with identity elements;
• semigroups—magmas where the operation is associative;
• monoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. In other words, it is a unital semigroup. Definition A monoid is a magma (M, ), i. a set M with binary ops—semigroups with identity elements;
• groups—monoids with inverse elementIn mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combinatis, or equivalently, associative quasigroups (which are always loops);
• abelian groupAbstract algebra Algebra Group theory In mathematics, an abelian group is a commutative group, i. a group G ) such that a b b a for all a and b in G''. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions fos—groups where the operation is commutative.

2 Free magma

A free magma on a set X is the "most general possible" magma generated by the set X (i.e. there are no relations or axioms imposed on the generators; see free objectThe idea of a free object in mathematics is one of the basics of abstract algebra. It is part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations); but on the other hand it has a clean formulati). It can be described, in terms familiar in computer scienceIn its most general sense, computer science CS or compsci is the study of computation and information processing, both in hardware and in software. Introduction Computer science encomposses a variety of topics relating to computation, ranging from abstrac, as the magma of full binary trees with leaves labelled by elements of X. The operation is that of joining trees at the root. It therefore has a foundational role in syntax.

See also: free semigroup, free group.