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| {{other uses|Magma (disambiguation)}}
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| In [[abstract algebra]], a '''magma''' (or '''groupoid'''; not to be confused with [[groupoid]]s in [[category theory]]) is a basic kind of [[algebraic structure]]. Specifically, a magma consists of a [[Set (mathematics)|set]] <math>M</math> equipped with a single [[binary operation]] <math>M \times M \rightarrow M</math>.
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| A binary operation is [[closure (binary operation)|closed]] by definition, but no other [[axiom]]s are imposed on the operation.
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| The term ''magma'' for this kind of structure was introduced by [[Nicolas Bourbaki]]. The term ''groupoid'' is an older, but still commonly used alternative which was introduced by [[Øystein Ore]].
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| {{Algebraic structures |Group}}
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| == Definition ==
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| A magma is a [[set (mathematics)|set]] <math>M</math> matched with an [[Binary operation|operation]] "<math>\cdot</math>" that sends any two [[element (mathematics)|elements]] <math>a,b \in M</math> to another element <math>a \cdot b</math>. The symbol "<math>\cdot</math>" is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation <math>(M,\cdot)</math> must satisfy the following requirement (known as the ''magma axiom''):
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| : For all <math>a</math>, <math>b</math> in <math>M</math>, the result of the operation <math>a \cdot b</math> is also in <math>M</math>.
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| And in mathematical notation:
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| : <math>\forall a,b \in M : a \cdot b \in M</math>
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| === Etymology ===
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| In French, the word "magma" has multiple common meanings, one of them being "jumble". It is likely that the French Bourbaki group referred to sets with well-defined binary operations as magmas with the "jumble" definition in mind.
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| == Types of magmas ==
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| Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation.
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| Commonly studied types of magmas include
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| * [[quasigroup]]s—[[nonempty]] magmas where [[division (mathematics)|division]] is always possible;
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| * [[loop (algebra)|loop]]s—quasigroups with [[identity element]]s;
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| * [[semigroup]]s—magmas where the operation is [[associative]];
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| * [[semilattice]]s—semigroups where the operation is [[commutative]] and [[idempotent]];
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| * [[monoid]]s—semigroups with [[identity element]]s;
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| * [[group (mathematics)|group]]s—monoids with [[inverse element]]s, or equivalently, associative loops or nonempty associative quasigroups;
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| * [[abelian group]]s—groups where the operation is commutative.
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| :[[Image:Magma to group2.svg|380px]]
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| ::''Note that both divisibility and invertibility''
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| ::''imply the [[cancellation property]].''
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| == Morphism of magmas ==
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| A [[morphism]] of magmas is a function <math>f\colon M\to N</math> mapping magma <math>M</math> to magma <math>N</math>, that preserves the binary operation:
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| :<math>f(x \; *_M \;y) = f(x) \; *_N\; f(y)</math>
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| where <math>*_M</math> and <math>*_N</math> denote the binary operation on <math>M</math> and <math>N</math> respectively.
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| ==Combinatorics and parentheses==
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| For the general, non-associative case, the magma operation may be repeatedly iterated. To denote pairings, parentheses are used. The resulting [[string (computer science)|string]] consists of symbols denoting elements of the magma, and balanced sets of parenthesis. The set of all possible strings of balanced parenthesis is called the [[Dyck language]]. The total number of different ways of writing <math>n</math> applications of the magma operator is given by the [[Catalan number]] <math>C_n</math>. Thus, for example, <math>C_2=2</math>, which is just the statement that <math>(ab)c</math> and <math>a(bc)</math> are the only two ways of pairing three elements of a magma with two operations. Less trivially, <math>C_3=5</math>: <math>((ab)c)d</math>, <math>(ab)(cd)</math>, <math>(a(bc))d</math>, <math>a((bc)d)</math>, and <math>a(b(cd))</math>.
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| A shorthand is often used to reduce the number of parentheses. This is accomplished by using juxtaposition in place of the operation. For example, if the magma operation is <math>*</math>, then <math>xy * z</math> abbreviates <math>(x * y) * z</math>. Further abbreviations are possible by inserting spaces, for example by writing <math>xy * z * wv</math> in place of <math>((x * y) * z) * (w * v)</math>. Of course, for more complex expressions the use of parenthesis turns out to be inevitable. A way to avoid completely the use of parentheses is [[prefix notation]].
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| == Free magma ==
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| A '''free magma''' <math>M_X</math> on a set <math>X</math> is the "most general possible" magma generated by the set <math>X</math> (i.e., there are no relations or axioms imposed on the generators; see [[free object]]). It can be described, in terms familiar in [[computer science]], as the magma of [[binary tree]]s with leaves labeled by elements of <math>X</math>. The operation is that of joining trees at the root. It therefore has a foundational role in [[syntax]].
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| A free magma has the [[universal property]] such that, if <math>f\colon X\to N</math> is a function from the set <math>X</math> to any magma <math>N</math>, then there is a unique extension of <math>f</math> to a morphism of magmas <math>f^\prime</math>
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| :<math>f^\prime\colon M_X \to N.</math>
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| ''See also'': [[free semigroup]], [[free group]], [[Hall set]], [[Wedderburn–Etherington number]]
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| == Classification by properties ==
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| {{Group-like structures}}
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| A magma (''S'', *) is called
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| * {{anchor|unital}}''[[unital algebra|unital]]'' if it has an identity element,
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| * ''[[medial magma|medial]]'' if it satisfies the identity ''xy'' * ''uz'' = ''xu'' * ''yz'' (i.e. (''x'' * ''y'') * (''u'' * ''z'') = (''x'' * ''u'') * (''y'' * ''z'') for all ''x'', ''y'', ''u'', ''z'' in ''S''),
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| * ''[[semimedial|left semimedial]]'' if it satisfies the identity ''xx'' * ''yz'' = ''xy'' * ''xz'',
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| * ''[[semimedial|right semimedial]]'' if it satisfies the identity ''yz'' * ''xx'' = ''yx'' * ''zx'',
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| * ''[[semimedial]]'' if it is both left and right semimedial,
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| * ''left distributive'' if it satisfies the identity ''x'' * ''yz'' = ''xy'' * ''xz'',
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| * ''right distributive'' if it satisfies the identity ''yz'' * ''x'' = ''yx'' * ''zx'',
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| * ''[[autodistributive]]'' if it is both left and right distributive,
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| * ''[[commutative]]'' if it satisfies the identity ''xy'' = ''yx'',
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| * ''[[idempotent]]'' if it satisfies the identity ''xx'' = ''x'',
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| * ''[[unipotent]]'' if it satisfies the identity ''xx'' = ''yy'',
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| * ''[[zeropotent]]'' if it satisfies the identity ''xx'' * ''y'' = ''yy'' * ''x'' = ''xx'',
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| * ''[[alternativity|alternative]]'' if it satisfies the identities ''xx'' * ''y'' = ''x'' * ''xy'' and ''x'' * ''yy'' = ''xy'' * ''y'',
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| * ''[[power-associative]]'' if the submagma generated by any element is associative,
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| * ''left-[[cancellative]]'' if for all ''x'', ''y'', and ''z'', ''xy'' = ''xz'' implies ''y'' = ''z''
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| * ''right-cancellative'' if for all ''x'', ''y'', and ''z'', ''yx'' = ''zx'' implies ''y'' = ''z''
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| * ''cancellative'' if it is both right-cancellative and left-cancellative
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| * a ''[[semigroup]]'' if it satisfies the identity ''x'' * ''yz'' = ''xy'' * ''z'' ([[associative|associativity]]),
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| * a ''[[Null_semigroup#Left_zero_semigroup|semigroup with left zeros]]'' if there are elements ''x'' for which the identity ''x'' = ''xy'' holds,
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| * a ''[[Null_semigroup#Right_zero_semigroup|semigroup with right zeros]]'' if there are elements ''x'' for which the identity ''x'' = ''yx'' holds,
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| * a ''semigroup with zero multiplication'' or a ''[[null semigroup]]'' if it satisfies the identity ''xy'' = ''uv'', for all x,y,u and v
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| * a ''left [[Unar (mathematics)|unar]]'' if it satisfies the identity ''xy'' = ''xz'',
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| * a ''right unar'' if it satisfies the identity ''yx'' = ''zx'',
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| * ''[[trimedial]]'' if any triple of its (not necessarily distinct) elements generates a medial submagma,
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| * ''[[entropic (algebra)|entropic]]'' if it is a [[universal algebra|homomorphic image]] of a medial [[cancellative|cancellation]] magma.
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| If <math>*</math> is instead a [[partial operation]], then ''S'' is called a '''partial magma'''.
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| ==Generalizations==
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| See [[n-ary group]].
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| ==See also==
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| *[[Magma category]]
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| *[[Auto magma object]]
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| *[[Universal algebra]]
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| *[[Magma computer algebra system]], named after the object of this article.
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| *An [[example of a commutative non-associative magma]]
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| *[[Algebraic_structure#Structures_whose_axioms_are_all_identities|Algebraic structures whose axioms are all identities]]
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| *[[Groupoid algebra]]
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| | [[Group (mathematics)|Group]]
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| | [[Monoid]]
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| | [[Semigroup]]
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| {|
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| | Magma
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| {| style="border-left:4px solid SkyBlue"
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| | [[Operation_(mathematics)|Operation]]
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| | [[Closure (mathematics)|Closure]]
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| |}
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| |}
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| |-
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| | [[Associativity]]
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| |}
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| |}
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| |-
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| | [[Identity element|Identity]]
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| |}
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| |}
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| |-
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| | [[Inverse element|Inverses]]
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| |}
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| |}
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| |}
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| |}
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| == References ==
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| <references />
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| * {{springer|id=m/m110040|author=M. Hazewinkel|title=Magma}}
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| * {{springer|id=f/f110190|author=M. Hazewinkel|title=Free magma}}
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| * {{mathworld|urlname=Groupoid|title=Groupoid}}
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| {{DEFAULTSORT:Magma (Algebra)}}
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| [[Category:Non-associative algebra]]
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| [[Category:Binary operations]]
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| [[Category:Algebraic structures]]
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