Magma category
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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 the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including
Clifford Clifford may refer to: People *Clifford (name), an English given name and surname, includes a list of people with that name *William Kingdon Clifford *Baron Clifford *Baron Clifford of Chudleigh *Baron de Clifford *Clifford baronets *Clifford fami ...
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 Boro ...
(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 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: : a, b \in M \implies a \cdot b \in M. 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, 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 . 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 ''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). The binary operation on ''MX'' is formed by wrapping each of the two operands in parenthesis and juxtaposing them in the same order. For example: : : : ''MX'' 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''′ : ''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 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 Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
. (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. ;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: 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: If it satisfies the identity ; Idempotent: If it satisfies the identity ; 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: If the submagma generated by any element is associative ; Flexible: if ;A semigroup, 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: 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 In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
of a medial cancellation magma.


Category of magmas

The category of magmas, denoted Mag, is the category 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 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 of the ( constant sequence of the) endomorphism. Because the singleton is the terminal object 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, 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