In

_{''M''} ''y'') = ''f''(''x'') •_{''N''} ''f''(''y'')
where •_{''M''} and •_{''N''} denote the binary operation on ''M'' and ''N'' respectively.

_{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 _{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 _{X}'' → ''N''.

abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

, a magma, binar or, rarely, groupoid is a basic kind of algebraic structure
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. Specifically, a magma consists of a set equipped with a single binary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

that must be closed by definition. No other properties are imposed.
History and terminology

The term ''groupoid'' was introduced in 1927 byHeinrich 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, a ...

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 (mathematics), group in several equivalent ways. A groupoid can be seen as a:
*''group (mat ...

(translated from the German ''Gruppoid''). The term was then appropriated by B. A. Hausmann and Øystein Ore
Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic group native to N ...

(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, formerly Zentralblatt MATH, is a major international reviewing service providing reviews and abstracts for articles in pure mathematics, pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for ...

, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a groupoid
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
Preston is a place name, surname and given name that may refer to:
Places
England
*Preston, Lancashire, an urban settlement
**The City of Preston, Lancashire, a borough and non-metropolitan district which contains the settlement
**County Borou ...

(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 ie Algebras and Lie Groups, 1965
Ie, ie, IE or I/E may refer to:
Arts and entertainment
*Iced Earth, a band from Florida, US
*Improv Everywhere, a comedy group
*Into Eternity (band), a band from Canada
*Individual events (speech), events centred around public speaking.
Busines ...

" 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 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: : $a,b\; \backslash in\; M\; \backslash implies\; a\; \backslash cdot\; b\; \backslash in\; M$. If • is instead apartial operation
In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands) to produce another element. More formally, a binary operation is an Operation (mathematics), operation of arity two.
More specif ...

, then is called a partial magma or more often a partial groupoidIn abstract algebra, a partial magma (algebra), groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial function, partial binary operation.
A partial groupoid is a partial algebra.
Partial semigroup
A parti ...

.
Morphism of magmas

Amorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

of magmas is a function, , mapping magma ''M'' to magma ''N'', that preserves the binary operation:
:''f'' (''x'' •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 isprefix 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
Operator may refer to:
Mathematics
* A ...

, in which the same expression would be written . Another way, familiar to programmers, is postfix notation Postfix may refer to:
* Postfix (linguistics), an affix which is placed after the stem of a word
* Postfix notation
Reverse Polish notation (RPN), also known as Polish postfix notation or simply postfix notation, is a mathematical notation in wh ...

(Reverse Polish notation
Reverse Polish notation (RPN), also known as Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in which operators ''precede'' their ...

), in which the same expression would be written , in which the order of execution is simply left-to-right (no Currying
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

).
The set of all possible string
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), ''Strings'' (1991 fil ...

s 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 (computer science)#Formal theory, string of square brackets

. The total number of different ways of writing applications of the magma operator is given by the and
And or AND may refer to:
Logic, grammar, and computing
* Conjunction (grammar)
In grammar
In linguistics
Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study o ...

The set of Dyck words forms the Dyck language.
Dyck words ...Catalan number
In combinatorial mathematics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is ...

, . 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, 4294967296, ... magmas with 0, 1, 2, 3, 4, ... elements. The corresponding numbers of non-isomorphic
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

magmas are 1, 1, 10, 3330, 178981952, ... and the numbers of simultaneously non-isomorphic and non-antiisomorphic
In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

magmas are 1, 1, 7, 1734, 89521056, ... .
Free magma

A free magma, ''Mfree object
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

). The binary operation on ''Mcomputer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , , and . Computer science ...

, as the magma of binary tree
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , ...

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
Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the ...

.
A free magma has the universal property
In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

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'' ′ : ''MTypes 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 (mathematics), group in the sense that "division (mathematics), division" is always possible. Quasigroups differ from groups mainly in that th ...

: 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

;Semigroup
In mathematics, a semigroup is an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

: A magma where the operation is associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

;Inverse semigroup In group (mathematics), 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'', ...

: A semigroup with inverse.
;Semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilatti ...

: A semigroup where the operation is commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and idempotent
Idempotence (, ) is the property of certain operations
Operation or Operations may refer to:
Science and technology
* Surgical operation
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, ...

;Monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

: A semigroup with an identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

;Group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

: A monoid with inverse element
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s, or equivalently, an associative loop, or a non-empty associative quasigroup
;Abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

: A group where the operation is commutative
Note that each of divisibility and invertibility imply the cancellation property
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

.
Classification by properties

A magma , with ∈ , is called ;Medial
Medial may refer to:
Mathematics
* Medial magma, a mathematical identity in algebra Geometry
* Medial axis, in geometry the set of all points having more than one closest point on an object's boundary
* Medial graph, another graph that repres ...

: 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

: If it satisfies the identity,
;Idempotent
Idempotence (, ) is the property of certain operations
Operation or Operations may refer to:
Science and technology
* Surgical operation
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, ...

: If it satisfies the identity,
;Unipotent
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

: If it satisfies the identity,
;Zeropotent: If it satisfies the identities,
;Alternative
''AlterNative: An International Journal of Indigenous Peoples'' (formerly ''AlterNative: An International Journal of Indigenous Scholarship'') is a quarterly peer-reviewed academic journal published by Ngā Pae o te Māramatanga, New Zealand’s ...

: If it satisfies the identities and
;Power-associative In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

: 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

, or associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

: 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 Semigroup#Identity and zero, zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then ...

: If it satisfies the identity,
;Unital: If it has an identity element
;Left-cancellative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

: If, for all , and, , implies
;Right-cancellative: If, for all , and, , implies
;Cancellative: If it is both right-cancellative and left-cancellative
;A semigroup with left zeros: If it is a semigroup and, for all , the identity, , holds
;A semigroup with right zeros: If it is a semigroup and, for all , the identity, , holds
;Trimedial: If any triple of (not necessarily distinct) elements generates a medial submagma
;Entropic: If it is a homomorphic image
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homom ...

of a medial cancellation magma.
Category of magmas

The category of magmas, denoted Mag, is thecategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

whose objects are magmas, and whose morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s are magma homomorphisms. The category Mag has direct products, and there is an inclusion functor
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

: as trivial magmas, with operations given by projection: .
An important property is that an injective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 group ...

can be extended to an automorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of a magma extension, 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 product (category theory), products, pullback (category theory), pullbacks and inverse limits. The du ...

of the (constant
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics)
In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...

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 group ...

.
Because the singleton is the terminal object
In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

of Mag, and because Mag is algebraic, Mag is pointed and complete
Complete may refer to:
Logic
* Completeness (logic)
* Complete theory, 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, ...

.
Generalizations

See ''n''-ary group.See also

* Magma category * Auto magma object *Universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spa ...

*Magma computer algebra system
Magma is a computer algebra system designed to solve problems in abstract algebra, algebra, number theory, algebraic geometry, geometry and combinatorics. It is named after the algebraic structure magma (algebra), magma. It runs on Unix-like operat ...

, named after the object of this article.
* Commutative non-associative magmas
* Algebraic structures whose axioms are all identities
*Groupoid algebra
*Hall set
References

* * * *Further reading

* {{DEFAULTSORT:Magma (Algebra) Non-associative algebra Binary operations Algebraic structures