Magma (algebra)

Inabstract algebra,amagma,binar,^[1]or, rarely,groupoidis a basic kind ofalgebraic structure.Specifically, a magma consists of asetequipped with a singlebinary operationthat must beclosedby definition. No other properties are imposed.

History and terminology

The termgroupoidwas introduced in 1927 byHeinrich Brandtdescribing hisBrandt groupoid.The term was then appropriated by B. A. Hausmann andØystein Ore(1937)^[2]in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers inZentralblatt,Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is agroupoidin the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, includingCliffordandPreston(1961) andHowie(1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the termgroupoidis "perhaps most often used in modern mathematics" in the sense given to it in category theory.^[3]

According to Bergman and Hausknecht (1996): "There is no generally accepted word for a set with a not necessarily associative binary operation. The wordgroupoidis 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 termmagmawas used bySerre[Lie Algebras and Lie Groups, 1965]. "^[4]It also appears inBourbaki'sÉléments de mathématique,Algèbre, chapitres 1 à 3, 1970.^[5]

Definition

A magma is asetMmatched with anoperation• that sends any twoelementsa,b∈Mto another element,a•b∈M.The symbol • is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation(M,•)must satisfy the following requirement (known as themagmaorclosure property):

For alla,binM,the result of the operationa•bis also inM.

And in mathematical notation:

a,b\in M\implies a\cdot b\in M.

If • is instead apartial operation,then(M,•)is called apartial magma^[6]or, more often, apartial groupoid.^[6]^[7]

Morphism of magmas

Amorphismof magmas is a functionf:M→Nthat maps magma(M,•)to magma(N,∗)that preserves the binary operation:

f(x•y) =f(x) ∗f(y).

For example, withMequal to thepositive real numbersand * as thegeometric mean,Nequal to the real number line, and • as thearithmetic mean,alogarithmfis a morphism of the magma (M,*) to (N,•).

proof:

\log {\sqrt {xy}}\ =\ {\frac {\log x+\log y}{2}}

Note that these commutative magmas are not associative; nor do they have anidentity element.This morphism of magmas has been used ineconomicssince 1863 whenW. Stanley Jevonscalculated the rate ofinflationin 39 commodities in England in hisA 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 • (b • c)) • d \equiv (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: $xy • z \equiv (x • y) • z$ .For example, the above is abbreviated to the following expression, still containing parentheses:

(a • bc) d

.

A way to avoid completely the use of parentheses isprefix notation,in which the same expression would be written $•• a • bcd$ .Another way, familiar to programmers, ispostfix notation(reverse Polish notation), in which the same expression would be written $abc •• d •$ ,in which the order of execution is simply left-to-right (nocurrying).

The set of all possiblestringsconsisting of symbols denoting elements of the magma, and sets of balanced parentheses is called theDyck language.The total number of different ways of writing $n$ applications of the magma operator is given by theCatalan number $C n$ .Thus, for example, $C 2 = 2$ ,which is just the statement that $(ab) c$ and $a (bc)$ are the only two ways of pairing three elements of a magma with two operations. Less trivially, $C 3 = 5$ : $((ab) c) d$ , $(a (bc)) d$ , $(ab)(cd)$ , $a ((bc) d)$ ,and $a (b (cd))$ .

There are $n n 2$ magmas with $n$ elements, so there are 1, 1, 16, 19683,4294967296,... (sequenceA002489in theOEIS) magmas with 0, 1, 2, 3, 4,... elements. The corresponding numbers of non-isomorphicmagmas are 1, 1, 10, 3330,178981952,... (sequenceA001329in theOEIS) and the numbers of simultaneously non-isomorphic and non-antiisomorphicmagmas are 1, 1, 7, 1734,89521056,... (sequenceA001424in theOEIS).^[8]

Free magma

Afree magmaM_Xon a setXis the "most general possible" magma generated byX(i.e., there are no relations or axioms imposed on the generators; seefree object). The binary operation onM_Xis formed by wrapping each of the two operands in parentheses and juxtaposing them in the same order. For example:

a • b = (a)(b),

a • (a • b) = (a)((a)(b)),

(a • a) • b = ((a)(a))(b).

M_Xcan be described as the set of non-associative words onXwith parentheses retained.^[9]

It can also be viewed, in terms familiar incomputer science,as the magma of fullbinary treeswith leaves labelled by elements ofX.The operation is that of joining trees at the root.

A free magma has theuniversal propertysuch that iff:X→Nis a function fromXto any magmaN,then there is a unique extension offto a morphism of magmasf′

f′:M_X→N.

Types of magma

Algebraic structures from magmas to groups

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 wheredivisionis always possible.
- Loop:A quasigroup with anidentity element.
Semigroup:A magma where the operation isassociative.
- Monoid:A semigroup with an identity element.
Group:A magma with inverse, associativity, and an identity element.

Note that each of divisibility and invertibility imply thecancellation property.

Magmas withcommutativity

Commutative magma:A magma with commutativity.
Commutative monoid:A monoid with commutativity.
Abelian group:A group with commutativity.

Classification by properties

Group-like structures
	Total	Associative	Identity	Divisible	Commutative
Partial magma	Unneeded	Unneeded	Unneeded	Unneeded	Unneeded
Semigroupoid	Unneeded	Required	Unneeded	Unneeded	Unneeded
Small category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded
Commutativegroupoid	Unneeded	Required	Required	Required	Required
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Commutativemagma	Required	Unneeded	Unneeded	Unneeded	Required
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Commutativequasigroup	Required	Unneeded	Unneeded	Required	Required
Unital magma	Required	Unneeded	Required	Unneeded	Unneeded
Commutativeunital magma	Required	Unneeded	Required	Unneeded	Required
Loop	Required	Unneeded	Required	Required	Unneeded
Commutativeloop	Required	Unneeded	Required	Required	Required
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Commutativesemigroup	Required	Required	Unneeded	Unneeded	Required
Associativequasigroup	Required	Required	Unneeded	Required	Unneeded
Commutative-and-associativequasigroup	Required	Required	Unneeded	Required	Required
Monoid	Required	Required	Required	Unneeded	Unneeded
Commutative monoid	Required	Required	Required	Unneeded	Required
Group	Required	Required	Required	Required	Unneeded
Abelian group	Required	Required	Required	Required	Required

A magma $(S,•)$ ,with $x, y, u, z$ ∈ $S$ ,is called

Medial: If it satisfies the identity $xy • uz \equiv xu • yz$
Left semimedial: If it satisfies the identity $xx • yz \equiv xy • xz$
Right semimedial: If it satisfies the identity $yz • xx \equiv yx • zx$
Semimedial: If it is both left and right semimedial
Left distributive: If it satisfies the identity $x • yz \equiv xy • xz$
Right distributive: If it satisfies the identity $yz • x \equiv yx • zx$
Autodistributive: If it is both left and right distributive
Commutative: If it satisfies the identity $xy \equiv yx$
Idempotent: If it satisfies the identity $xx \equiv x$
Unipotent: If it satisfies the identity $xx \equiv yy$
Zeropotent: If it satisfies the identities $xx • y \equiv xx \equiv y • xx$ ^[10]
Alternative: If it satisfies the identities $xx • y \equiv x • xy$ and $x • yy \equiv xy • y$
Power-associative: If the submagma generated by any element is associative
Flexible: if $xy • x \equiv x • yx$
Associative: If it satisfies the identity $x • yz \equiv xy • z$ ,called asemigroup
A left unar: If it satisfies the identity $xy \equiv xz$
A right unar: If it satisfies the identity $yx \equiv zx$
Semigroup with zero multiplication, ornull semigroup: If it satisfies the identity $xy \equiv uv$
Unital: If it has an identity element
Left-cancellative: If, for all $x, y, z$ ,relation $xy = xz$ implies $y = z$
Right-cancellative: If, for all $x, y, z$ ,relation $yx = zx$ implies $y = z$
Cancellative: If it is both right-cancellative and left-cancellative
Asemigroup with left zeros: If it is a semigroup and it satisfies the identity $xy \equiv x$
Asemigroup with right zeros: If it is a semigroup and it satisfies the identity $yx \equiv x$
Trimedial: If any triple of (not necessarily distinct) elements generates a medial submagma
Entropic: If it is ahomomorphic imageof a medialcancellationmagma.^[11]
Central: If it satisfies the identity $xy • yz \equiv y$

Number of magmas satisfying given properties

Idempotence	Commutative property	Associative property	Cancellation property	OEIS sequence (labeled)	OEIS sequence (isomorphism classes)
Unneeded	Unneeded	Unneeded	Unneeded	A002489	A001329
Required	Unneeded	Unneeded	Unneeded	A090588	A030247
Unneeded	Required	Unneeded	Unneeded	A023813	A001425
Unneeded	Unneeded	Required	Unneeded	A023814	A001423
Unneeded	Unneeded	Unneeded	Required	A002860add a(0)=1	A057991
Required	Required	Unneeded	Unneeded	A076113	A030257
Required	Unneeded	Required	Unneeded
Required	Unneeded	Unneeded	Required
Unneeded	Required	Required	Unneeded	A023815	A001426
Unneeded	Required	Unneeded	Required		A057992
Unneeded	Unneeded	Required	Required	A034383add a(0)=1	A000001with a(0)=1 instead of 0
Required	Required	Required	Unneeded
Required	Required	Unneeded	Required		a(n)=1 for n=0 and all odd n, a(n)=0 for all even n≥2
Required	Unneeded	Required	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
Unneeded	Required	Required	Required	A034382add a(0)=1	A000688add a(0)=1
Required	Required	Required	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, denotedMag,is thecategorywhose objects are magmas and whosemorphismsaremagma homomorphisms.The categoryMaghasdirect products,and there is aninclusion functor:Set→ Med ↪ Magas trivial magmas, withoperationsgiven byprojection $x T y = y$ .

An important property is that aninjective endomorphismcan be extended to anautomorphismof a magmaextension,just thecolimitof the (constantsequence of the)endomorphism.

Because thesingleton $({*}, *)$ is theterminal objectofMag,and becauseMagisalgebraic,Magis pointed andcomplete.^[12]

References

^Bergman, Clifford (2011),Universal Algebra: Fundamentals and Selected Topics,CRC Press,ISBN 978-1-4398-5130-2
^Hausmann, B. A.; Ore, Øystein (October 1937), "Theory of quasi-groups",American Journal of Mathematics,59(4): 983–1004,doi:10.2307/2371362,JSTOR 2371362.
^Hollings, Christopher (2014),Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups,American Mathematical Society, pp. 142–143,ISBN 978-1-4704-1493-1.
^Bergman, George M.; Hausknecht, Adam O. (1996),Cogroups and Co-rings in Categories of Associative Rings,American Mathematical Society, p. 61,ISBN 978-0-8218-0495-7.
^Bourbaki, N. (1998) [1970],"Algebraic Structures: §1.1 Laws of Composition: Definition 1",Algebra I: Chapters 1–3,Springer, p. 1,ISBN 978-3-540-64243-5.
^^a ^bMüller-Hoissen, Folkert; Pallo, Jean Marcel; Stasheff, Jim, eds. (2012),Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift,Springer, p. 11,ISBN 978-3-0348-0405-9.
^Evseev, A. E. (1988), "A survey of partial groupoids", in Silver, Ben (ed.),Nineteen Papers on Algebraic Semigroups,American Mathematical Society,ISBN 0-8218-3115-1.
^Weisstein, Eric W."Groupoid".MathWorld.
^Rowen, Louis Halle (2008),"Definition 21B.1.",Graduate Algebra: Noncommutative View,Graduate Studies in Mathematics,American Mathematical Society,p. 321,ISBN 978-0-8218-8408-9.
^Kepka, T.; Němec, P. (1996),"Simple balanced groupoids"(PDF),Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica,35(1): 53–60.
^ Ježek, Jaroslav; Kepka, Tomáš (1981),"Free entropic groupoids"(PDF),Commentationes Mathematicae Universitatis Carolinae,22(2): 223–233,MR 0620359.
^Borceux, Francis; Bourn, Dominique (2004).Mal'cev, protomodular, homological and semi-abelian categories.Springer. pp. 7, 19.ISBN 1-4020-1961-0.

Hazewinkel, M. (2001) [1994],"Magma",Encyclopedia of Mathematics,EMS Press
Hazewinkel, M. (2001) [1994],"Groupoid",Encyclopedia of Mathematics,EMS Press
Hazewinkel, M. (2001) [1994],"Free magma",Encyclopedia of Mathematics,EMS Press
Weisstein, Eric W."Groupoid".MathWorld.