Magma (algebra)

Template:Short description Template:About Template:Algebraic structures In abstract algebra, a magma, binar,<ref>Template:Citation</ref> 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.

History and terminologyEdit

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)<ref>Template:Citation.</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 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.<ref name="Hollings2014">Template:Citation.</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 Serre [Lie Algebras and Lie Groups, 1965]."<ref name="BergmanHausknecht1996">Template:Citation.</ref> It also appears in Bourbaki's {{#invoke:Lang|lang}}.<ref name="Bourbaki1998">Template:Citation.</ref>

DefinitionEdit

A magma is a set M matched with an operation • that sends any two elements Template:Nowrap to another element, Template:Nowrap. The symbol • is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation Template:Nowrap must satisfy the following requirement (known as the magma or closure property):

For all a, b in M, the result of the operation Template:Nowrap is also in M.

And in mathematical notation:

<math>a, b \in M \implies a \cdot b \in M.</math>

If • is instead a partial operation, then Template:Nowrap is called a partial magma<ref name="Müller-HoissenPallo2012">Template:Citation.</ref> or, more often, a partial groupoid.<ref name="Müller-HoissenPallo2012"/><ref name="Silver">Template:Citation.</ref>

Morphism of magmasEdit

A morphism of magmas is a function Template:Nowrap that maps magma Template:Nowrap to magma Template:Nowrap 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: <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 combinatoricsEdit

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:

Template:Math.

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: Template:Math. For example, the above is abbreviated to the following expression, still containing parentheses:

Template:Math.

A way to avoid completely the use of parentheses is prefix notation, in which the same expression would be written Template:Math. Another way, familiar to programmers, is postfix notation (reverse Polish notation), in which the same expression would be written Template:Math, in which the order of execution is simply left-to-right (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 Template:Math applications of the magma operator is given by the Catalan number Template:Math. Thus, for example, Template:Math, which is just the statement that Template:Math and Template:Math are the only two ways of pairing three elements of a magma with two operations. Less trivially, Template:Math: Template:Math, Template:Math, Template:Math, Template:Math, and Template:Math.

There are Template:Math magmas with Template:Math elements, so there are 1, 1, 16, 19683, Template:Val, ... (sequence A002489 in the OEIS) magmas with 0, 1, 2, 3, 4, ... elements. The corresponding numbers of non-isomorphic magmas are 1, 1, 10, 3330, Template:Val, ... (sequence A001329 in the OEIS) and the numbers of simultaneously non-isomorphic and non-antiisomorphic magmas are 1, 1, 7, 1734, Template:Val, ... (sequence A001424 in the OEIS).<ref>Template:Mathworld</ref>

Free magmaEdit

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:

Template:Math

M_X can be described as the set of non-associative words on X with parentheses retained.<ref>Template:Citation.</ref>

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.

A free magma has the universal property such that if Template:Nowrap is a function from X to any magma N, then there is a unique extension of f to a morphism of magmas Template:Itco Template:Prime

Template:Itco Template:Prime : M_X → N.

Template:See also

Types of magmaEdit

File:Magma to group4.svg

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 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 propertiesEdit

Template:Group-like structures A magma Template:Math, with Template:Math ∈ Template:Math, is called

Medial: If it satisfies the identity Template:Math
Left semimedial: If it satisfies the identity Template:Math
Right semimedial: If it satisfies the identity Template:Math
Semimedial: If it is both left and right semimedial
Left distributive: If it satisfies the identity Template:Math
Right distributive: If it satisfies the identity Template:Math
Autodistributive: If it is both left and right distributive
Commutative: If it satisfies the identity Template:Math
Idempotent: If it satisfies the identity Template:Math
Unipotent: If it satisfies the identity Template:Math
Zeropotent: If it satisfies the identities Template:Math<ref>Template:Citation.</ref>
Alternative: If it satisfies the identities Template:Math and Template:Math
Power-associative: If the submagma generated by any element is associative
Flexible: if Template:Math
Associative: If it satisfies the identity Template:Math, called a semigroup
A left unar: If it satisfies the identity Template:Math
A right unar: If it satisfies the identity Template:Math
Semigroup with zero multiplication, or null semigroup: If it satisfies the identity Template:Math
Template:AnchorUnital: If it has an identity element
Left-cancellative: If, for all Template:Math, relation Template:Math implies Template:Math
Right-cancellative: If, for all Template:Math, relation Template:Math implies Template:Math
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 Template:Math
A semigroup with right zeros: If it is a semigroup and it satisfies the identity Template:Math
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.<ref>

Template:Citation.</ref>

Central: If it satisfies the identity Template:Math

Number of magmas satisfying given propertiesEdit

Idempotence	Commutative property	Associative property	Cancellation property	OEIS sequence (labeled)	OEIS sequence (isomorphism classes)
Template:No	Template:No	Template:No	Template:No	A002489	A001329
Template:Yes	Template:No	Template:No	Template:No	A090588	A030247
Template:No	Template:Yes	Template:No	Template:No	A023813	A001425
Template:No	Template:No	Template:Yes	Template:No	A023814	A001423
Template:No	Template:No	Template:No	Template:Yes	A002860 add a(0)=1	A057991
Template:Yes	Template:Yes	Template:No	Template:No	A076113	A030257
Template:Yes	Template:No	Template:Yes	Template:No
Template:Yes	Template:No	Template:No	Template:Yes
Template:No	Template:Yes	Template:Yes	Template:No	A023815	A001426
Template:No	Template:Yes	Template:No	Template:Yes		A057992
Template:No	Template:No	Template:Yes	Template:Yes	A034383 add a(0)=1	A000001 with a(0)=1 instead of 0
Template:Yes	Template:Yes	Template:Yes	Template:No
Template:Yes	Template:Yes	Template:No	Template:Yes		a(n)=1 for n=0 and all odd n, a(n)=0 for all even n≥2
Template:Yes	Template:No	Template:Yes	Template:Yes	a(0)=a(1)=1, a(n)=0 for all n≥2	a(0)=a(1)=1, a(n)=0 for all n≥2
Template:No	Template:Yes	Template:Yes	Template:Yes	A034382 add a(0)=1	A000688 add a(0)=1
Template:Yes	Template:Yes	Template:Yes	Template:Yes	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 magmasEdit

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: Template:Nobr as trivial magmas, with operations given by projection Template:Math. More generally, because Mag is algebraic, it is a complete category.<ref>Template:Cite book</ref>

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.

ReferencesEdit

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