Magma (mathematics) | Wikipedia audio article

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 equipped with a single binary operation. The binary
operation must be closed by definition but no other properties are imposed.==History and terminology==
The term groupoid was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid
(translated from the German Gruppoid). The term was then appropriated by B. A. Hausmann
and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this
article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed
with this overloading of terminology. The Brandt groupoid is a groupoid in the sense
used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential
books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid
in the sense of Hausmann and Ore. Hollings (2014) writes that the term groupoid is “perhaps
most often used in modern mathematics” in the sense given to it in category theory.According
to Bergman and Hausknecht (1996): “There is no generally accepted word for a set with
a not necessarily associative binary operation. The word groupoid is used by many universal
algebraists, but workers in category theory and related areas object strongly to this
usage because they use the same word to mean ‘category in which all morphisms are invertible’.
The term magma was used by Serre [Lie Algebras and Lie Groups, 1965].” It also appears in
Bourbaki’s Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970.==Definition==
A magma is a set M matched with an operation, •, that sends any two elements a, b ∈ M
to another element, a • b. The symbol, •, is a general placeholder for a properly defined
operation. To qualify as a magma, the set and operation (M, •) must satisfy the following
requirement (known as the magma or closure axiom): For all a, b in M, the result of the operation
a • b is also in M.And in mathematical notation: a
, b
∈ M ⟹ a
⋅ b
∈ M {\displaystyle a,b\in M\implies a\cdot b\in
M} .If • is instead a partial operation, then
S is called a partial magma or more often a partial groupoid.==Morphism of magmas==
A morphism of magmas is a function, f : M → 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.==Notation and combinatorics==
The magma operation may be applied repeatedly, and in the general, non-associative case,
the order matters, which is notated with parentheses. Also, the operation, •, is often omitted
and notated by juxtaposition: (a • (b • c)) • d=(a(bc))dA shorthand
is often used to reduce the number of parentheses, in which the innermost operations and pairs
of parentheses are omitted, being replaced just with juxtaposition, xy • z=(x • y)
• z. For example, the above is abbreviated to the following expression, still containing
parentheses: (a • bc)d.A way to avoid completely the
use of parentheses is prefix notation, in which the same expression would be written
••a•bcd. Another way, familiar to programmers, is postfix notation (Reverse Polish notation),
in which the same expression would be written abc••d•, in which the order of execution
is simply left-to-right (no Currying). The set of all possible strings consisting
of symbols denoting elements of the magma, and sets of balanced parentheses 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, Cn. Thus, for example, C2=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, C3=5: ((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), and
a(b(cd)). There are n n 2 {\displaystyle n^{n^{2}}}
magmas with n {\displaystyle n}
elements so 1, 1, 16, 19683, 4294967296, … (sequence A002489 in the OEIS) magmas with 0, 1, 2,
3, 4, … elements. The corresponding numbers of non-isomorphic magmas are 1, 1, 10, 3330,
178981952, … (sequence A001329 in the OEIS) and of simultaneously non-isomorphic and non-antiisomorphic
magmas are 1, 1, 7, 1734, 89521056, … (sequence A001424 in the OEIS).==Free magma==
A free magma, MX, on a set, X, is the “most general possible” magma generated by X (i.e.,
there are no relations or axioms imposed on the generators; see free object). It can be
described as the set of non-associative words on X with parentheses retained.It can also
be viewed, in terms familiar in computer science, as the magma of 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. A free magma has the universal property such
that, if f : X → N is a function from X to any magma, N, then there is a unique extension
of f to a morphism of magmas, f ′ f ′ : MX → N.==Types of magma==Magmas are not often studied as such; instead
there are several different kinds of magma, depending on what axioms the operation is
required to satisfy. Commonly studied types of magma include: Quasigroup
A magma where division is always possible Loop
A quasigroup with an identity element Semigroup
A magma where the operation is associative Semilattice
A semigroup where the operation is commutative and idempotent
Monoid A semigroup with an identity element
Group A monoid with inverse elements, or equivalently,
an associative loop, or a non-empty associative quasigroup
Abelian group A group where the operation is commutativeNote
that each of divisibility and invertibility imply the cancellation property.==Classification by properties==
A magma (S, •), with x, y, u, z ∈ S, is called Medial
If it satisfies the identity, xy • uz ≡ xu • yz
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 identities, xx • y ≡ xx
≡ y • 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, or associative
If it satisfies the identity, x • yz ≡ xy • z
A left unar If it satisfies the identity, xy ≡ xz
A right unar If it satisfies the identity, yx ≡ zx
Semigroup with zero multiplication, or null semigroup
If it satisfies the identity, xy ≡ uv Unital
If it has an identity element 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 with left zeros
If it is a semigroup and, for all x, the identity, x ≡ xy, holds
A semigroup with right zeros If it is a semigroup and, for all x, the identity,
x ≡ yx, holds Trimedial
If any triple of (not necessarily distinct) elements generates a medial submagma
Entropic If it is a homomorphic image of a medial cancellation
See n-ary group.==See also==
Magma category Auto magma object
Universal algebra Magma computer algebra system, named after
the object of this article. Commutative non-associative magmas
Algebraic structures whose axioms are all identities
Groupoid algebra

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