View source for Magma (algebra)

{{Short description|Algebraic structure with a binary operation}}
{{About|the algebraic structure|groupoids in category theory|Groupoid|other uses|Magma (disambiguation)}}
{{Algebraic structures |Group}}
In [[abstract algebra]], a '''magma''', '''binar''',<ref>{{citation |first=Clifford |last=Bergman|title=Universal Algebra: Fundamentals and Selected Topics |publisher=CRC Press |date=2011 |isbn=978-1-4398-5130-2 |url={{GBurl|snvRBQAAQBAJ|pg=PR7}}}}</ref> or, rarely, '''groupoid''' is a basic kind of [[algebraic structure]]. Specifically, a magma consists of a [[Set (mathematics)|set]] equipped with a single [[binary operation]] that must be [[closure (binary operation)|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.&nbsp;A. Hausmann and [[Øystein Ore]] (1937)<ref>{{citation |first1=B. A. |last1=Hausmann |first2=Øystein |last2=Ore |title=Theory of quasi-groups |journal=American Journal of Mathematics |volume=59 |issue=4 |pages=983–1004 |date=October 1937 |jstor=2371362 |doi=10.2307/2371362}}.</ref> 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 the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including [[Alfred Hoblitzelle Clifford|Clifford]] and [[G. B. Preston|Preston]] (1961) and [[John Mackintosh Howie|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.<ref name="Hollings2014">{{citation |first=Christopher |last=Hollings |title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups |url=https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA142 |year=2014 |publisher=American Mathematical Society |isbn=978-1-4704-1493-1 |pages=142–143}}.</ref>

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 [[Jean-Pierre Serre|Serre]] [Lie Algebras and Lie Groups, 1965]."<ref name="BergmanHausknecht1996">{{citation |first1=George M. |last1=Bergman |first2=Adam O. |last2=Hausknecht |title=Cogroups and Co-rings in Categories of Associative Rings |url=https://books.google.com/books?id=s6NnkQs3JBMC&pg=PA61 |year=1996 |publisher=American Mathematical Society |isbn=978-0-8218-0495-7 |page=61}}.</ref> It also appears in [[Nicolas Bourbaki|Bourbaki]]'s {{lang|fr|[[Éléments de mathématique]], Algèbre, chapitres 1 à 3, 1970}}.<ref name="Bourbaki1998">{{citation |first=N. |last=Bourbaki |title=Algebra I: Chapters 1–3 |chapter=Algebraic Structures: §1.1 Laws of Composition: Definition 1 |chapter-url=https://books.google.com/books?id=STS9aZ6F204C&pg=PA1 |year=1998 |publisher=Springer |isbn=978-3-540-64243-5 |page=1 |orig-year=1970}}.</ref>

== Definition ==

A magma is a [[set (mathematics)|set]] ''M'' with an [[Binary operation|operation]] • that sends any two [[element (mathematics)|elements]] {{nowrap|''a'', ''b'' ∈ ''M''}} to another element, {{nowrap|''a'' • ''b'' ∈ ''M''}}. The symbol • is a general placeholder for a properly defined operation. This requirement that for all ''a'', ''b'' in ''M'', the result of the operation {{nowrap|''a'' • ''b''}} also be in ''M'', is known as the ''magma'' or [[closure property]]. In mathematical notation:
: <math>\forall a,b\colon a, b \in M \implies a \cdot b \in M.</math>

If • is instead a [[partial operation]], then {{nowrap|(''M'', •)}} is called a '''partial magma'''<ref name="Müller-HoissenPallo2012">{{citation |editor-first=Folkert |editor-last=Müller-Hoissen |editor2-first=Jean Marcel |editor2-last=Pallo |editor3-first=Jim |editor3-last=Stasheff |title=Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift |url=https://books.google.com/books?id=Y01d6g5UemQC&pg=PA11 |year=2012 |publisher=Springer |isbn=978-3-0348-0405-9 |page=11}}.</ref> or, more often, a [[partial groupoid]].<ref name="Müller-HoissenPallo2012"/><ref name="Silver">{{citation |editor-first=Ben |editor-last=Silver |title=Nineteen Papers on Algebraic Semigroups |publisher=American Mathematical Society |isbn=0-8218-3115-1 |last=Evseev |first=A. E. |chapter=A survey of partial groupoids |year=1988}}.</ref>

== Morphism of magmas ==

A [[morphism]] of magmas is a function {{nowrap|''f'' : ''M'' → ''N''}} that maps a magma {{nowrap|(''M'', •)}} to a magma {{nowrap|(''N'', ∗)}} and preserves the binary operation:
: ''f'' (''x'' • ''y'') = ''f''(''x'') ∗ ''f''(''y'').

For example, with ''M'' equal to the [[positive real numbers]] and • as the [[geometric mean]], ''N'' equal to the real number line, and ∗ as the [[arithmetic mean]], a [[logarithm]] ''f'' is a morphism of the magma (''M'', •) to (''N'', ∗).
:proof: <math>\log{\sqrt{x y} } \ = \ \frac{\log x + \log y}{2}. </math>
Note that these commutative magmas are not associative; nor do they have an [[identity element]]. This morphism of magmas has been used in [[economics]] since 1863 when [[W. Stanley Jevons]] calculated the rate of [[inflation]] 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:
: {{math|1= (''a'' • (''b'' • ''c'')) • ''d'' ≡ (''a''(''bc''))''d''}}.

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: {{math|1=''xy'' • ''z'' ≡ (''x'' • ''y'') • ''z''}}. For example, the above is abbreviated to the following expression, still containing parentheses:
: {{math|1= (''a'' • ''bc'')''d''}}.
A way to avoid completely the use of parentheses is [[prefix notation]], in which the same expression would be written {{math|1= ••''a''•''bcd''}}. Another way, familiar to programmers, is [[postfix notation]] ([[reverse Polish notation]]), in which the same expression would be written {{math|1= ''abc''••''d''•}}, in which the order of execution is simply left-to-right (no [[currying]]).

The set of all possible [[string (computer science)|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 {{math|''n''}} applications of the magma operator is given by the [[Catalan number]] {{math|''C<sub>n</sub>''}}. Thus, for example, {{math|1=''C''<sub>2</sub> = 2}}, which is just the statement that {{math|(''ab'')''c''}} and {{math|''a''(''bc'')}} are the only two ways of pairing three elements of a magma with two operations. Less trivially, {{math|1=''C''<sub>3</sub> = 5}}: {{math|((''ab'')''c'')''d''}}, {{math|(''a''(''bc''))''d''}}, {{math|(''ab'')(''cd'')}}, {{math|''a''((''bc'')''d'')}}, and {{math|''a''(''b''(''cd''))}}.

There are {{math|''n''<sup>''n''<sup>2</sup></sup>}} magmas with {{math|''n''}} elements, so there are 1, 1, 16, 19683, {{val|4294967296}},&nbsp;... {{OEIS|A002489}} magmas with 0, 1, 2, 3, 4,&nbsp;... elements. The corresponding numbers of non-[[isomorphic]] magmas are 1, 1, 10, 3330, {{val|178981952}},&nbsp;... {{OEIS|A001329}} and the numbers of simultaneously non-isomorphic and non-[[antiisomorphic]] magmas are 1, 1, 7, 1734, {{val|89521056}},&nbsp;... {{OEIS|A001424}}.<ref>{{mathworld|urlname=Groupoid|title=Groupoid}}</ref>

== Free magma ==

A '''free magma''' ''M<sub>X</sub>'' 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 ''M<sub>X</sub>'' is formed by wrapping each of the two operands in parentheses and juxtaposing them in the same order. For example:
: {{math|1= ''a'' • ''b'' = (''a'')(''b''),}}
: {{math|1= ''a'' • (''a'' • ''b'') = (''a'')((''a'')(''b'')),}}
: {{math|1= (''a'' • ''a'') • ''b'' = ((''a'')(''a''))(''b'').}}

''M<sub>X</sub>'' can be described as the set of non-associative words on ''X'' with parentheses retained.<ref>{{citation | title=Graduate Algebra: Noncommutative View | page=321 | series=[[Graduate Studies in Mathematics]] | first=Louis Halle | last=Rowen | publisher=[[American Mathematical Society]] | year=2008 | isbn=978-0-8218-8408-9 |chapter=Definition 21B.1. |chapter-url=https://books.google.com/books?id=8svFC09gGeMC&pg=PA321 }}.</ref>

It can also be viewed, in terms familiar in [[computer science]], as the magma of full [[binary tree]]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]] that if {{nowrap|''f'' : ''X'' → ''N''}} is a function from ''X'' to any magma ''N'', then there is a unique extension of ''f'' to a morphism of magmas {{itco|''f''}}{{prime}}
: {{itco|''f''}}{{prime}} : ''M<sub>X</sub>'' → ''N''.

{{see also|Free semigroup|Free group|Hall set|Wedderburn–Etherington number}}

== Types of magma ==
{{Dark mode invert|[[File:Magma to group2.svg|thumb|Diagram showing the relationships between different algebraic structures]]}}
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 (mathematics)|division]] is always possible.
** [[Loop (algebra)|'''Loop''']]: A quasigroup with an [[identity element]].
* '''[[Semigroup]]''': A magma where the operation is [[associative]].
** '''[[Monoid]]''': A semigroup with an identity element.
* [[Group (mathematics)|'''Group''']]: A magma with inverse, associativity, and an identity element.

Note that each of divisibility and invertibility imply the [[cancellation property]].

; Magmas with [[commutative|commutativity]]:
* '''[[Commutative magma]]''': A magma with commutativity.
* '''[[Commutative monoid]]''': A monoid with commutativity.
* '''[[Abelian group]]''': A group with commutativity.

== Classification by properties ==
{{Group-like structures}}
A magma {{math|(''S'', •)}}, with {{math|''x'', ''y'', ''u'', ''z''}} ∈ {{math|''S''}}, is called

;[[Medial magma|Medial]]: If it satisfies the identity {{math|''xy'' • ''uz'' ≡ ''xu'' • ''yz''}}
;Left semimedial: If it satisfies the identity {{math|''xx'' • ''yz'' ≡ ''xy'' • ''xz''}}
;Right semimedial: If it satisfies the identity {{math|''yz'' • ''xx'' ≡ ''yx'' • ''zx''}}
;Semimedial: If it is both left and right semimedial
;Left distributive: If it satisfies the identity {{math|''x'' • ''yz'' ≡ ''xy'' • ''xz''}}
;Right distributive: If it satisfies the identity {{math|''yz'' • ''x'' ≡ ''yx'' • ''zx''}}
;Autodistributive: If it is both left and right distributive
;[[Commutative magma|Commutative]]: If it satisfies the identity {{math|''xy'' ≡ ''yx''}}
;[[Idempotent]]: If it satisfies the identity {{math|''xx'' ≡ ''x''}}
;Unipotent: If it satisfies the identity {{math|''xx'' ≡ ''yy''}}
;Zeropotent: If it satisfies the identities {{math|''xx'' • ''y'' ≡ ''xx'' ≡ ''y'' • ''xx''}}<ref>{{citation |first1=T. |last1=Kepka |first2=P. |last2=Němec |title=Simple balanced groupoids |journal=Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |volume=35 |issue=1 |pages=53–60 |year=1996 |url=http://dml.cz/bitstream/handle/10338.dmlcz/120353/ActaOlom_35-1996-1_7.pdf }}.</ref>
;[[Alternativity|Alternative]]: If it satisfies the identities {{math|''xx'' • ''y'' ≡ ''x'' • ''xy''}} and {{math|''x'' • ''yy'' ≡ ''xy'' • ''y''}}
;[[Power-associative]]: If the submagma generated by any element is associative
;[[Flexible algebra|Flexible]]: if {{math|''xy'' • ''x'' ≡ ''x'' • ''yx''}}
;[[Associative]]: If it satisfies the identity {{math|''x'' • ''yz'' ≡ ''xy'' • ''z''}}, called a [[semigroup]]
;A left unar: If it satisfies the identity {{math|''xy'' ≡ ''xz''}}
;A right unar: If it satisfies the identity {{math|''yx'' ≡ ''zx''}}
;Semigroup with zero multiplication, or [[null semigroup]]: If it satisfies the identity {{math|''xy'' ≡ ''uv''}}
;{{anchor|unital}}Unital: If it has an identity element
;Left-[[cancellative]]: If, for all {{math|''x'', ''y'', ''z''}}, relation {{math|''xy'' {{=}} ''xz''}} implies {{math|''y'' {{=}} ''z''}}
;Right-cancellative: If, for all {{math|''x'', ''y'', ''z''}}, relation {{math|''yx'' {{=}} ''zx''}} implies {{math|''y'' {{=}} ''z''}}
;Cancellative: If it is both right-cancellative and left-cancellative
;A [[Null semigroup#Left zero semigroup|semigroup with left zeros]]: If it is a semigroup and it satisfies the identity {{math|''xy'' ≡ ''x''}}
;A [[Null semigroup#Right zero semigroup|semigroup with right zeros]]: If it is a semigroup and it satisfies the identity {{math|''yx'' ≡ ''x''}}
;Trimedial: If any triple of (not necessarily distinct) elements generates a medial submagma
;Entropic: If it is a [[universal algebra|homomorphic image]] of a medial [[cancellative|cancellation]] magma.<ref>
{{citation
 | last1 = Ježek | first1 = Jaroslav
 | last2 = Kepka | first2 = Tomáš
 | issue = 2
 | journal = Commentationes Mathematicae Universitatis Carolinae
 | mr = 620359
 | pages = 223–233
 | title = Free entropic groupoids
 | url = http://dml.cz/bitstream/handle/10338.dmlcz/106070/CommentatMathUnivCarol_022-1981-2_2.pdf
 | volume = 22
 | year = 1981
}}.</ref>
;[[Central groupoid|Central]]: If it satisfies the identity {{math|''xy'' • ''yz'' ≡ ''y''}}

== Number of magmas satisfying given properties ==

{|class=wikitable
![[Idempotence]]
![[Commutative property]]
![[Associative property]]
![[Cancellation property]]
!OEIS sequence (labeled)
!OEIS sequence (isomorphism classes)
|-
|{{no|Unneeded}}||{{no|Unneeded}}||{{no|Unneeded}}||{{no|Unneeded}}||{{OEIS link|A002489}}||{{OEIS link|A001329}}
|-
|{{yes|Required}}||{{no|Unneeded}}||{{no|Unneeded}}||{{no|Unneeded}}||{{OEIS link|A090588}}||{{OEIS link|A030247}}
|-
|{{no|Unneeded}}||{{yes|Required}}||{{no|Unneeded}}||{{no|Unneeded}}||{{OEIS link|A023813}}||{{OEIS link|A001425}}
|-
|{{no|Unneeded}}||{{no|Unneeded}}||{{yes|Required}}||{{no|Unneeded}}||{{OEIS link|A023814}}||{{OEIS link|A001423}}
|-
|{{no|Unneeded}}||{{no|Unneeded}}||{{no|Unneeded}}||{{yes|Required}}||{{OEIS link|A002860}} add a(0)=1||{{OEIS link|A057991}}
|-
|{{yes|Required}}||{{yes|Required}}||{{no|Unneeded}}||{{no|Unneeded}}||{{OEIS link|A076113}}||{{OEIS link|A030257}}
|-
|{{yes|Required}}||{{no|Unneeded}}||{{yes|Required}}||{{no|Unneeded}}||||
|-
|{{yes|Required}}||{{no|Unneeded}}||{{no|Unneeded}}||{{yes|Required}}||||
|-
|{{no|Unneeded}}||{{yes|Required}}||{{yes|Required}}||{{no|Unneeded}}||{{OEIS link|A023815}}||{{OEIS link|A001426}}
|-
|{{no|Unneeded}}||{{yes|Required}}||{{no|Unneeded}}||{{yes|Required}}||||{{OEIS link|A057992}}
|-
|{{no|Unneeded}}||{{no|Unneeded}}||{{yes|Required}}||{{yes|Required}}||{{OEIS link|A034383}} add a(0)=1||{{OEIS link|A000001}} with a(0)=1 instead of 0
|-
|{{yes|Required}}||{{yes|Required}}||{{yes|Required}}||{{no|Unneeded}}||||
|-
|{{yes|Required}}||{{yes|Required}}||{{no|Unneeded}}||{{yes|Required}}||||a(n)=1 for n=0 and all odd n, a(n)=0 for all even n≥2
|-
|{{yes|Required}}||{{no|Unneeded}}||{{yes|Required}}||{{yes|Required}}||a(0)=a(1)=1, a(n)=0 for all n≥2||a(0)=a(1)=1, a(n)=0 for all n≥2
|-
|{{no|Unneeded}}||{{yes|Required}}||{{yes|Required}}||{{yes|Required}}||{{OEIS link|A034382}} add a(0)=1||{{OEIS link|A000688}} add a(0)=1
|-
|{{yes|Required}}||{{yes|Required}}||{{yes|Required}}||{{yes|Required}}||a(0)=a(1)=1, a(n)=0 for all n≥2||a(0)=a(1)=1, a(n)=0 for all n≥2
|}

== Category of magmas ==
The category of magmas, denoted '''Mag''', is the [[Category (mathematics)|category]] whose objects are magmas and whose [[morphism]]s are [[Magma (algebra)#Morphism of magmas|magma homomorphisms]]. The category '''Mag''' has [[Product (category theory)|direct products]], and there is an [[inclusion functor]]: {{nobr|'''[[category of sets|Set]] ↪ Mag'''}} as trivial magmas, with [[binary operation|operation]]s given by [[Projection (mathematics)|projection]] {{math|1=''x'' T ''y'' = ''y'' }}. More generally, because '''Mag''' is [[algebraic category|algebraic]], it is a [[complete category]].<ref>{{Cite book |title=Mal'cev, protomodular, homological and semi-abelian categories |last1=Borceux |first1=Francis |last2=Bourn |first2=Dominique |publisher=Springer |year=2004 |pages=7, 19 |isbn=1-4020-1961-0|url=https://books.google.com/books?id=olmeksCI22cC&pg=PA19}}</ref>

An important property is that an [[injective]] [[endomorphism]] can be extended to an [[automorphism]] of a magma [[algebraic extension|extension]], just the [[colimit]] of the ([[constant function|constant]] sequence of the) [[endomorphism]].

== See also ==
* [[Universal algebra]]
* [[Magma computer algebra system]], named after the object of this article.
* [[Commutative magma]]
* [[Algebraic structure#Structures whose axioms are all identities|Algebraic structures whose axioms are all identities]]
* [[Groupoid algebra]]
* [[Hall set]]

== References ==
{{reflist}}
{{refbegin}}
* {{springer |id=m/m110040|first=M. |last=Hazewinkel |title=Magma}}
* {{springer |id=Groupoid|first=M. |last=Hazewinkel |title=Groupoid}}
* {{springer |id=f/f110190|first=M. |last=Hazewinkel |title=Free magma}}
* {{mathworld |urlname=Groupoid |title=Groupoid}}
{{refend}}

== Further reading ==
* {{citation |first=Richard Hubert |last=Bruck |author-link=Richard Hubert Bruck |title=A survey of binary systems |year=1971 |publisher=Springer |isbn=978-0-387-03497-3 |edition=3rd}}

{{Authority control}}

{{DEFAULTSORT:Magma (Algebra)}}
[[Category:Non-associative algebra]]
[[Category:Binary operations]]
[[Category:Algebraic structures]]