magma (algebra)

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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), 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 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, 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: : $a,b \in M \implies a \cdot b \in M$. If • is instead a
partial 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

A
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 ...

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 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
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 ...
. The total number of different ways of writing applications of the magma operator is given by the
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, ''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 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 ''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 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'' ′ : ''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 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 the
category 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.

* 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

* * * *