In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, a magma, binar, or, rarely, groupoid is a basic kind of
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
. Specifically, a magma consists of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
equipped with a single
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
that must be
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
by definition. No other properties are imposed.
History and terminology
The term ''groupoid'' was introduced in 1927 by
Heinrich Brandt
Heinrich Brandt (8 November 1886, in Feudingen – 9 October 1954, in Halle, Saxony-Anhalt) was a German mathematician who was the first to develop the concept of a groupoid.
Brandt studied at the University of Göttingen and, from 1910 to 1913, ...
describing his
Brandt groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*''Group'' with a partial func ...
(translated from the German ). The term was then appropriated by B. A. Hausmann and
Øystein Ore
Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics.
Life
Ore graduated from the University of Oslo in 1922, with a ...
(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
zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ...
, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*''Group'' with a partial functi ...
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
Howie is a Scottish locational surname derived from a medieval estate in Ayrshire, southwest Scotland. While its ancient name is known as "The lands of How", its exact location is lost to time. The word "How", predating written history, appears ...
(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 ie Algebras and Lie Groups, 1965"
[.] It also appears in
Bourbaki's .
[.]
Definition
A magma is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''M'' matched with an
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
• that sends any two
elements to another element, . The symbol • is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation must satisfy the following requirement (known as the ''magma'' or ''closure axiom''):
: For all ''a'', ''b'' in ''M'', the result of the operation is also in ''M''.
And in mathematical notation:
:
If • is instead a
partial operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary ope ...
, then is called a partial magma
[.] or more often a
partial groupoid
In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation.
A partial groupoid is a partial algebra.
Partial semigroup
A partial groupoid (G,\circ) is called ...
.
[.]
Morphism of magmas
A
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
of magmas is a function 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 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: . For example, the above is abbreviated to the following expression, still containing parentheses:
:
A way to avoid completely the use of parentheses is
prefix notation
Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators ''precede'' their operands, in contrast t ...
, in which the same expression would be written . Another way, familiar to programmers, is
postfix notation
Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in whi ...
(
reverse Polish notation
Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in whi ...
), in which the same expression would be written , in which the order of execution is simply left-to-right (no
currying
In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f that ...
).
The set of all possible
strings
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian anim ...
consisting of symbols denoting elements of the magma, and sets of balanced parentheses is called the
Dyck language
In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of square brackets and The set of Dyck words forms the Dyck language.
Dyck words and language are named after the mathematici ...
. The total number of different ways of writing applications of the magma operator is given by the
Catalan number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Cata ...
. Thus, for example, , which is just the statement that and are the only two ways of pairing three elements of a magma with two operations. Less trivially, : , , , , and .
There are magmas with elements, so there are 1, 1, 16, 19683, , ... magmas with 0, 1, 2, 3, 4, ... elements. The corresponding numbers of non-
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
magmas are 1, 1, 10, 3330, , ... and the numbers of simultaneously non-isomorphic and non-
antiisomorphic
In category theory, a branch of mathematics, an antiisomorphism (or anti-isomorphism) between structured sets ''A'' and ''B'' is an isomorphism from ''A'' to the opposite of ''B'' (or equivalently from the opposite of ''A'' to ''B''). If there ...
magmas are 1, 1, 7, 1734, , ... .
Free magma
A free magma ''M
X'' 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
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between ele ...
). The binary operation on ''M
X'' is formed by wrapping each of the two operands in parenthesis and juxtaposing them in the same order. For example:
:
:
:
''M
X'' 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
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, as the magma of
binary tree
In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...
s 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
In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituency) ...
.
A free magma has the
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
such that if is a function from ''X'' to any magma ''N'', then there is a unique extension of ''f'' to a morphism of magmas ''f''′
: ''f''′ : ''M
X'' → ''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
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have ...
: A magma where
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
is always possible.
**
Loop: A quasigroup with an
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
.
*
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
: A magma where the operation is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
.
**
Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
: A semigroup with an identity element.
*
Inverse semigroup In group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', i.e. a regular semig ...
: A semigroup with
inverse. (Also a quasigroup with associativity)
*
Group: A magma with inverse, associativity, and an identity element.
Note that each of divisibility and invertibility imply the
cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that .
An ...
.
;Magmas with
commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
:
*
Commutative magma
In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras.
A magma which is b ...
: A magma with commutativity.
*
Semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a mee ...
: A monoid with commutativity.
*
Abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
: A group with commutativity.
Classification by properties
A magma , with ∈ , is called
;
Medial: If it satisfies the identity
;Left semimedial: If it satisfies the identity
;Right semimedial: If it satisfies the identity
;Semimedial: If it is both left and right semimedial
;Left distributive: If it satisfies the identity
;Right distributive: If it satisfies the identity
;Autodistributive: If it is both left and right distributive
;
Commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
: If it satisfies the identity
;
Idempotent
Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...
: If it satisfies the identity
;
Unipotent
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''.
In particular, a square matrix ''M'' is a unipotent ...
: If it satisfies the identity
;Zeropotent: If it satisfies the identities
;
Alternative
Alternative or alternate may refer to:
Arts, entertainment and media
* Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki''
* ''The Alternative'' (film), a 1978 Australian television film
* ''The Alternative ...
: If it satisfies the identities and
;
Power-associative In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
Definition
An algebra (or more generally a magma) is said to be power-associative if the subalgebra ge ...
: If the submagma generated by any element is associative
;
Flexible
Flexible may refer to:
Science and technology
* Power cord, a flexible electrical cable.
** Flexible cable, an Electrical cable as used on electrical appliances
* Flexible electronics
* Flexible response
* Flexible-fuel vehicle
* Flexible rake rec ...
: if
;A
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
, or
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
: If it satisfies the identity
;A left unar: If it satisfies the identity
;A right unar: If it satisfies the identity
;Semigroup with zero multiplication, or
null semigroup In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then the semigroup is called a le ...
: If it satisfies the identity
;Unital: If it has an identity element
;Left-
cancellative
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that .
A ...
: If, for all , relation implies
;Right-cancellative: If, for all , relation implies
;Cancellative: If it is both right-cancellative and left-cancellative
;A
semigroup with left zeros: If it is a semigroup and it satisfies the identity
;A
semigroup with right zeros: If it is a semigroup and it satisfies the identity
;Trimedial: If any triple of (not necessarily distinct) elements generates a medial submagma
;Entropic: If it is a
homomorphic image of a medial
cancellation magma.
Category of magmas
The category of magmas, denoted Mag, is the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
whose objects are magmas and whose
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s are
magma homomorphisms. The category Mag has
direct products, and there is an
inclusion functor
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...
: as trivial magmas, with
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
s given by
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
.
An important property is that an
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
can be extended to an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
of a magma
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
, just the
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
of the (
constant sequence of the)
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
.
Because the
singleton
Singleton may refer to:
Sciences, technology Mathematics
* Singleton (mathematics), a set with exactly one element
* Singleton field, used in conformal field theory Computing
* Singleton pattern, a design pattern that allows only one instance ...
is the
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
of Mag, and because Mag is
algebraic, Mag is pointed and
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
.
See also
*
Magma category
*
Universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, ...
*
Magma computer algebra system
Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma. It runs on Unix-like operating systems, as well as Windows.
Introduction
Magma ...
, named after the object of this article.
*
Commutative magma
In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras.
A magma which is b ...
*
Algebraic structures whose axioms are all identities
*
Groupoid algebra In mathematics, the concept of groupoid algebra generalizes the notion of group ring, group algebra.
Definition
Given a groupoid (G, \cdot) (in the sense of a category theory, category with all arrows invertible) and a field (mathematics), field ...
*
Hall set
In mathematics, in the areas of group theory and combinatorics, Hall words provide a unique monoid factorisation of the free monoid. They are also totally ordered, and thus provide a total order on the monoid. This is analogous to the better-known ...
References
* .
* .
* .
*
Further reading
*
{{DEFAULTSORT:Magma (Algebra)
Non-associative algebra
Binary operations
Algebraic structures