Magma (algebra)

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In abstract algebra, a magma (or groupoid) is a 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 closed by definition, but no other axioms are imposed on the operation.

The term magma for this kind of structure was introduced by Bourbaki. The term groupoid is an older, but still commonly used alternative which was introduced by Øystein Ore. However, groupoid also refers to an entirely different algebraic structure described at groupoid.

1 Types of magmas
2 Morphism of magmas
3 Combinatorics and parentheses
4 Free magma
5 More definitions
6 See also
7 References

[edit] 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;
monoids—semigroups with identity elements;
groups—monoids with inverse elements, or equivalently, associative loops (which are always quasigroups);
abelian groups—groups where the operation is commutative.

From magma to group, via two alternative paths. Key:

M = magma, d = divisibility, a = associativity,

Q = quasigroup, S = semigroup, e = identity.

L = loop, i = inversibility, N = monoid, G = group

Note that both divisibility and inversibility imply

the existence of the cancellation property.

[edit] Morphism of magmas

A morphism of magmas is a function $f:M\to N$ mapping magma $M$ to magma $N$ , that preserves the binary operation:

$f(x \; *_M \;y) = f(x) \; *_N\; f(y)$

where $* M$ and $* N$ denote the binary operation on $M$ , respectively on $N$ .

[edit] Combinatorics and parentheses

For the general, non-associative case, the magma operation may be repeatedly iterated. To denote pairings, parentheses are used. The resulting string consists of symbols denoting elements of the magma, and balanced sets of parenthesis. The set of all possible strings of balanced parenthesis is called the Dyck language. The total number of different ways of writing n applications of the magma operator is given by the Catalan number $C n$ . Thus, for example, $C 2 = 2$ , which is just the statement that $(a b) c$ and $a (b c)$ are the only two ways of pairing three elements of a magma with two operations.

A shorthand is often used in order to avoid as much parenthesis as possible. This is accomplished by using juxtaposition in place of the operation. For example, if the magma operation is *, then xy*z abbreviates (x * y) * z. Further abbreviations are possible by inserting spaces, for example by writing xy*z * wv in place of ((x * y) * z) * (w * v). Of course, for more complex expressions the use of parenthesis turns out to be inevitable. A way to avoid completely the use of parentheses is prefix notation, which is, however, counterintuitive.

[edit] Free magma

A free magma $M X$ 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 binary trees with leaves labeled by elements of X. The operation is that of joining trees at the root. It therefore has a foundational role in syntax.

A free magma has the universal property such that, if $f:X\to N$ is a function from the set X to any magma N, then there is a unique extension of $f$ to a morphism of magmas $f^\prime$

$f^\prime:M_X \to N.$

See also: free semigroup, free group, Hall set

[edit] More definitions

	Totality	Associativity	Identity	Division
Group-like structures
Group	Yes	Yes	Yes	Yes
Monoid	Yes	Yes	Yes	No
Semigroup	Yes	Yes	No	No
Loop	Yes	No	Yes	Yes
Quasigroup	Yes	No	No	Yes
Magma	Yes	No	No	No
Groupoid	No	Yes	Yes	Yes
Category	No	Yes	Yes	No

A magma (S, *) is called

unital if it has an identity element,
medial if it satisfies the identity xy * uz = xu * yz (i.e. (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,
left-cancellative if for all x, y, and z, xy = xz implies y = z
right-cancellative if for all x, y, and z, yx = zx implies y = z
cancellative if it is both right-cancellative and left-cancellative
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.