Magma (algebra)

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In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set $M$ equipped with a single binary operation $M \times M \rightarrow M$.

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 Nicolas Bourbaki. The term groupoid is an older, but still commonly used alternative which was introduced by Øystein Ore.

Definition

A magma is a set $M$ matched with an operation "$\cdot$" that sends any two elements $a,b \in M$ to another element $a \cdot b$. The symbol "$\cdot$" is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation $(M,\cdot)$ must satisfy the following requirement (known as the magma axiom):

For all $a$, $b$ in $M$, the result of the operation $a \cdot b$ is also in $M$.

And in mathematical notation:

$\forall a,b \in M : a \cdot b \in M$

Etymology

In French, the word "magma" has multiple common meanings, one of them being "jumble". It is likely that the French Bourbaki group referred to sets with well-defined binary operations as magmas with the "jumble" definition in mind.

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

Note that both divisibility and invertibility
imply the cancellation property.

Morphism of magmas

A morphism of magmas is a function $f\colon M\to N$ 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 $M$ and $N$ respectively.

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 $(ab)c$ and $a(bc)$ are the only two ways of pairing three elements of a magma with two operations. Less trivially, $C_3=5$: $((ab)c)d$, $(ab)(cd)$, $(a(bc))d$, $a((bc)d)$, and $a(b(cd))$.

A shorthand is often used to reduce the number of parentheses. 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.

Free magma

A free magma $M_X$ 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 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 $f\colon X\to N$ is a function from the set $X$ to any magma $N$, then there is a unique extension of $f$ to a morphism of magmas $f^\prime$

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

Classification by properties

 Totality* Associativity Identity Inverses Group-like structures Yes No No No No Yes Yes No No No Yes Yes Yes No No Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes No Yes Yes** No Yes No No No No No Yes Yes Yes No No Yes Yes No No No Yes No No No *Closure, which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently. **Each element of a loop has a left and right inverse, but these need not coincide.

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 there are elements x for which the identity x = xy holds,
• a semigroup with right zeros if there are elements x for which the identity x = yx holds,
• a semigroup with zero multiplication or a null semigroup if it satisfies the identity xy = uv, for all x,y,u and v
• 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.

If $*$ is instead a partial operation, then S is called a partial magma.

See n-ary group.