• 107328  Infos

# Magma (algebra)

In abstract algebra a '''magma''' (also called a '''groupoid''') is a particularly basic kind of algebraic structure Specifically a magma consists of a set ''M'' equipped with a single binary operation ''M'' × ''M'' → ''M'' A binary operation is closure (binary operation)|closed by definition but no other axioms are imposed on the operation
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

## 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 operationCommonly studied types of magmas include
• quasigroups—nonempty magmas where division is always possible;
• loop—quasigroups with identity elements;
• semigroups—magmas where the operation is associative;
• monoids—semigroups with identity elements;
• group—monoids with inverse elements or equivalently associative quasigroups (which are always loops);
• abelian groups—groups where the operation is commutative

## Free magma

A free magma on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object) It can be described in terms familiar in computer science 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

## More definitions

A magma (S *) is called
• medial if it satisfies the identity xy * uz = xu * yz (ie (x * y) * (u * z) = (x * u) * (y * z) for all x y u z in S)
• left semimedial if it satisfies the identity xx * yz = xy * xz
• right semimedial if it satisfies the identity yz * xx = yx * zx
• semimedial if it is both left and right semimedial
• left distributive if it satisfies the identity x * yz = xy * xz
• right distributive if it satisfies the identity yz * x = yx * zx
• autodistributive if it is both left and right distributive
• commutative if it satisfies the identity xy = yx
• idempotent if it satisfies the identity xx = x
• unipotent if it satisfies the identity xx = yy
• zeropotent if it satisfies the identity xx * y = yy * x = xx
• alternative if it satisfies the identities xx * y = x * xy and x * yy = xy * y
• power-associative if the submagma generated by any element is associative
• a semigroup if it satisfies the identity x * yz = xy * z (associativity)
• a semigroup with left zeros if it satisfies the identity x = xy
• a semigroup with right zeros if it satisfies the identity x = yx
• a semigroup with zero multiplication if it satisfies the identity xy = uv
• a left unar if it satisfies the identity xy = xz
• a right unar if it satisfies the identity yx = zx
• trimedial if any triple of its (not necessarily distinct) elements generates a medial submagma
• entropic if it is a homomorphic image of a medial cancellation magma