Algebraic structures 

In abstract algebra, a magma, binar,^{[1]} or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed.
YouTube Encyclopedic

1/5Views:1 24020 7489171 307503

Fractals in a Magma Algebra

MORE OCTONION FRACTALS

Räkna ut arean av en rektangel

Fractals in Several Groupoid or Semigroup Algebras

Procentenheter och procent
Transcription
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)^{[2]} 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 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.^{[3]}
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 [Lie Algebras and Lie Groups, 1965]."^{[4]} It also appears in Bourbaki's Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970.^{[5]}
Definition
A magma is a set M matched with an operation • that sends any two elements a, b ∈ M to 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 the magma or closure property):
 For all a, b in M, the result of the operation a • b is also in M.
And in mathematical notation:
If • is instead a partial operation, then (M, •) is called a partial magma^{[6]} or, more often, a partial groupoid.^{[6]}^{[7]}
Morphism of magmas
A morphism of magmas is a function f : M → N that maps magma (M, •) to magma (N, ∗) that 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:
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, nonassociative case, the order matters, which is notated with parentheses. Also, the operation • is often omitted and notated by juxtaposition:
 (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: xy • z ≡ (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 is prefix notation, in which the same expression would be written ••a•bcd. Another way, familiar to programmers, is postfix notation (reverse Polish notation), in which the same expression would be written abc••d•, in which the order of execution is simply lefttoright (no currying).
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 n applications of the magma operator is given by the Catalan 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^{n2} magmas with n elements, so there are 1, 1, 16, 19683, 4294967296, ... (sequence A002489 in the OEIS) magmas with 0, 1, 2, 3, 4, ... elements. The corresponding numbers of nonisomorphic magmas are 1, 1, 10, 3330, 178981952, ... (sequence A001329 in the OEIS) and the numbers of simultaneously nonisomorphic and nonantiisomorphic magmas are 1, 1, 7, 1734, 89521056, ... (sequence A001424 in the OEIS).^{[8]}
Free magma
A free magma M_{X} 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_{X} is 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_{X} can be described as the set of nonassociative words on X with parentheses retained.^{[9]}
It can also be viewed, in terms familiar in computer science, as the magma of full binary trees 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.
A free magma has the universal property such that if 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 f′
 f′ : M_{X} → 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: A magma where division is always possible.
 Loop: A quasigroup with an identity element.
 Semigroup: A magma where the operation is associative.
 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 the cancellation property.
 Magmas with commutativity
 Commutative magma: A magma with commutativity.
 Commutative monoid: A monoid with commutativity.
 Abelian group: A group with commutativity.
Classification by properties
Closure  Associative  Identity  Cancellation  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 
Commutative Groupoid  Unneeded  Required  Required  Required  Required 
Magma  Required  Unneeded  Unneeded  Unneeded  Unneeded 
Commutative magma  Required  Unneeded  Unneeded  Unneeded  Required 
Quasigroup  Required  Unneeded  Unneeded  Required  Unneeded 
Commutative quasigroup  Required  Unneeded  Unneeded  Required  Required 
Unital magma  Required  Unneeded  Required  Unneeded  Unneeded 
Commutative unital magma  Required  Unneeded  Required  Unneeded  Required 
Loop  Required  Unneeded  Required  Required  Unneeded 
Commutative loop  Required  Unneeded  Required  Required  Required 
Semigroup  Required  Required  Unneeded  Unneeded  Unneeded 
Commutative semigroup  Required  Required  Unneeded  Unneeded  Required 
Associative quasigroup  Required  Required  Unneeded  Required  Unneeded 
Commutativeandassociative quasigroup  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 ≡ xu • yz
 Left semimedial
 If it satisfies the identity xx • yz ≡ xy • xz
 Right semimedial
 If it satisfies the identity yz • xx ≡ yx • zx
 Semimedial
 If it is both left and right semimedial
 Left distributive
 If it satisfies the identity x • yz ≡ xy • xz
 Right distributive
 If it satisfies the identity yz • x ≡ yx • zx
 Autodistributive
 If it is both left and right distributive
 Commutative
 If it satisfies the identity xy ≡ yx
 Idempotent
 If it satisfies the identity xx ≡ x
 Unipotent
 If it satisfies the identity xx ≡ yy
 Zeropotent
 If it satisfies the identities xx • y ≡ xx ≡ y • xx^{[10]}
 Alternative
 If it satisfies the identities xx • y ≡ x • xy and x • yy ≡ xy • y
 Powerassociative
 If the submagma generated by any element is associative
 Flexible
 if xy • x ≡ x • yx
 Associative
 If it satisfies the identity x • yz ≡ xy • z, called a semigroup
 A left unar
 If it satisfies the identity xy ≡ xz
 A right unar
 If it satisfies the identity yx ≡ zx
 Semigroup with zero multiplication, or null semigroup
 If it satisfies the identity xy ≡ uv
 Unital
 If it has an identity element
 Leftcancellative
 If, for all x, y, z, relation xy = xz implies y = z
 Rightcancellative
 If, for all x, y, z, relation yx = zx implies y = z
 Cancellative
 If it is both rightcancellative and leftcancellative
 A semigroup with left zeros
 If it is a semigroup and it satisfies the identity xy ≡ x
 A semigroup with right zeros
 If it is a semigroup and it satisfies the identity yx ≡ x
 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.^{[11]}
 Central
 If it satisfies the identity xy • yz ≡ 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  A002860 add 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  A034383 add a(0)=1  A000001 with 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  A034382 add a(0)=1  A000688 add 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, denoted Mag, is the category whose objects are magmas and whose morphisms are magma homomorphisms. The category Mag has direct products, and there is an inclusion functor: Set → Med ↪ Mag as trivial magmas, with operations given by projection x T y = y .
An important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the (constant sequence of the) endomorphism.
Because the singleton ({*}, *) is the terminal object of Mag, and because Mag is algebraic, Mag is pointed and complete.^{[12]}
See also
 Magma category
 Universal algebra
 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
 ^ Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics, CRC Press, ISBN 9781439851302
 ^ Hausmann, B. A.; Ore, Øystein (October 1937), "Theory of quasigroups", 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 9781470414931.
 ^ Bergman, George M.; Hausknecht, Adam O. (1996), Cogroups and Corings in Categories of Associative Rings, American Mathematical Society, p. 61, ISBN 9780821804957.
 ^ Bourbaki, N. (1998) [1970], "Algebraic Structures: §1.1 Laws of Composition: Definition 1", Algebra I: Chapters 1–3, Springer, p. 1, ISBN 9783540642435.
 ^ ^{a} ^{b} MüllerHoissen, Folkert; Pallo, Jean Marcel; Stasheff, Jim, eds. (2012), Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift, Springer, p. 11, ISBN 9783034804059.
 ^ Evseev, A. E. (1988), "A survey of partial groupoids", in Silver, Ben (ed.), Nineteen Papers on Algebraic Semigroups, American Mathematical Society, ISBN 0821831151.
 ^ 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 9780821884089.
 ^ 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 semiabelian categories. Springer. pp. 7, 19. ISBN 1402019610.
 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.
Further reading
 Bruck, Richard Hubert (1971), A survey of binary systems (3rd ed.), Springer, ISBN 9780387034973