Magma (algebra)
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In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a magma, binar, or, rarely, groupoid is a basic kind of
algebraic structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
. 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, a binary operation ...
that must be closed by definition. No other properties are imposed.


History and terminology

The term ''groupoid'' was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid. The term was then appropriated by B. A. Hausmann and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a
groupoid In 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 fu ...
in the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term ''groupoid'' is "perhaps most often used in modern mathematics" in the sense given to it in category theory.. According to Bergman and Hausknecht (1996): "There is no generally accepted word for a set with a not necessarily associative binary operation. The word ''groupoid'' is used by many universal algebraists, but workers in category theory and related areas object strongly to this usage because they use the same word to mean 'category in which all morphisms are invertible'. The term ''magma'' was used by Serre 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'' 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 property): : 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, a binary operation o ...
, then is called a partial magma. or, more often, a partial groupoid..


Morphism of magmas

A
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
of magmas is a function that maps magma to magma that preserves the binary operation: : ''f'' (''x'' • ''y'') = ''f''(''x'') ∗ ''f''(''y''). For example, with ''M'' equal to the
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
and • as the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
, ''N'' equal to the real number line, and ∗ as the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
, a
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
''f'' is a morphism of the magma (''M'', •) to (''N'', ∗). :proof: \log \ = \ \frac Note that these commutative magmas are not associative; nor do they have an
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
. This morphism of magmas has been used in
economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...
since 1863 when W. Stanley Jevons calculated the rate of
inflation In economics, inflation is an increase in the average price of goods and services in terms of money. This increase is measured using a price index, typically a consumer price index (CPI). When the general price level rises, each unit of curre ...
in 39 commodities in England in his ''A Serious Fall in the Value of Gold Ascertained'', page 7.


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, Eastern Notation or simply prefix notation, is a mathematical notation in which operators ''precede'' their oper ...
, 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 Operation (mathematics), operators ''follow'' their operands, in contrast to pr ...
(
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 prefix or Polish notation ...
), 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 a function that takes multiple arguments into a sequence of families of functions, each taking a single argument. In the prototypical example, one begins with a functi ...
). The set of all possible strings consisting of symbols denoting elements of the magma, and sets of balanced parentheses is called the Dyck language. The total number of different ways of writing applications of the magma operator is given by the
Catalan number The Catalan numbers are a sequence of natural numbers that occur in various Enumeration, counting problems, often involving recursion, recursively defined objects. They are named after Eugène Charles Catalan, Eugène Catalan, though they were p ...
. 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 or morphism 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 the ...
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 parentheses 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, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, as the magma of full
binary tree In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theor ...
s with leaves labelled by elements of ''X''. The operation is that of joining trees at the root. 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 : : ''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 that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ...
: A magma where division is always possible. ** Loop: A quasigroup with an
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
. *
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 (just notation, not necessarily th ...
: A magma where the operation is
associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
. **
Monoid In abstract algebra, 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 . Monoids are semigroups with identity ...
: A semigroup with an identity element. * 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. Perhaps most familiar as a p ...
: * Commutative magma: A magma with commutativity. *
Commutative monoid In abstract algebra, 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 . Monoids are semigroups with identity ...
: 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 commu ...
: 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. Perhaps most familiar as a pr ...
: If it satisfies the identity ;
Idempotent Idempotence (, ) is the property of certain 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 arises in a number of pl ...
: If it satisfies the identity ;
Unipotent In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' âˆ’ 1 is a nilpotent element; in other words, (''r'' âˆ’ 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipote ...
: If it satisfies the identity ;Zeropotent: If it satisfies the identities ; Alternative: If it satisfies the identities and ; Power-associative: If the submagma generated by any element is associative ; Flexible: if ;
Associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
: If it satisfies the identity , called 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 (just notation, not necessarily th ...
;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 ...
: If it satisfies the identity ;Unital: If it has an identity element ;Left- cancellative: 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. ; Central: If it satisfies the identity


Number of magmas satisfying given properties


Category of magmas

The category of magmas, denoted Mag, is the
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
whose objects are magmas and whose
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s are magma homomorphisms. The category Mag has direct products, and there is an inclusion functor: as trivial magmas, with operations given by projection . More generally, because Mag is algebraic, it is a
complete category In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in ...
. 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 of its codomain; that is, implies (equivalently by contraposition, impl ...
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 automorphism ...
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 products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions s ...
of the ( constant sequence of the) endomorphism.


See also

*
Universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of stud ...
* Magma computer algebra system, named after the object of this article. * Commutative magma * Algebraic structures whose axioms are all identities * Groupoid algebra * Hall set


References

* * * *


Further reading

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