abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, the term associator is used in different ways as a measure of the non-associativity of an

algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...

. Associators are commonly studied as

triple system In algebra, a triple system (or ternar) is a vector space ''V'' over a field F together with a F-trilinear map : (\cdot,\cdot,\cdot) \colon V\times V \times V\to V. The most important examples are Lie triple systems and Jordan triple systems. The ...

Ring theory

For a

non-associative ring A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

''R'', the associator is the

multilinear map Multilinear may refer to: * Multilinear form, a type of mathematical function from a vector space to the underlying field * Multilinear map, a type of mathematical function between vector spaces * Multilinear algebra, a field of mathematics ...

: R \times R \times R \to R

given by :

,y,z = (xy)z - x(yz).

Just as the

commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...

, y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

= xy - yx measures the degree of non-commutativity, the associator measures the degree of non-associativity of ''R''. For an associative ring or

the associator is identically zero. The associator in any ring obeys the identity :

w,y,z +,x,y = x,y,z -,xy,z +,x,yz

The associator is alternating precisely when ''R'' is an alternative ring. The associator is symmetric in its two rightmost arguments when ''R'' is a pre-Lie algebra. The nucleus is the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of elements that associate with all others: that is, the ''n'' in ''R'' such that :

,R,R =,n,R =,R,n = \ \ .

The nucleus is an associative subring of ''R''.

Quasigroup theory

quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ...

''Q'' is a set with a

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...

\cdot : Q \times Q \to Q

such that for each ''a'', ''b'' in ''Q'', the equations

a \cdot x = b

and

y \cdot a = b

have unique solutions ''x'', ''y'' in ''Q''. In a quasigroup ''Q'', the associator is the map

(\cdot,\cdot,\cdot) : Q \times Q \times Q \to Q

defined by the equation :

(a\cdot b)\cdot c = (a\cdot (b\cdot c))\cdot (a,b,c)

for all ''a'', ''b'', ''c'' in ''Q''. As with its ring theory analog, the quasigroup associator is a measure of nonassociativity of ''Q''.

Higher-dimensional algebra

higher-dimensional algebra In mathematics, especially (Higher category theory, higher) category theory, higher-dimensional algebra is the study of Categorification, categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebr ...

, where there may be non-identity

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

s between algebraic expressions, an associator is an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

a_ : (xy)z \mapsto x(yz).

Category theory

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, the associator expresses the associative properties of the internal product

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

monoidal categories In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left an ...

References

* * Non-associative algebra {{algebra-stub

Ring theory

Quasigroup theory

Higher-dimensional algebra

Category theory

See also

References