abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

, the term associator is used in different ways as a measure of the

non-associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

of an

algebraic structure In mathematics, an algebraic structure 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 multiplication), and a finite set of ...

. 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 one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

R

, the associator is the

multilinear map In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W ar ...

: 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, a ...

, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

= xy - yx measures the degree of

non-commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

, the associator measures the degree of non-associativity of

R

. For an

associative ring In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying pro ...

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 In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is o ...

. The associator is symmetric in its two rightmost arguments when

R

is a

pre-Lie algebra In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as Tree (graph theory), rooted trees and vector fields on affine space. The notion of pre-Lie algebra has been introduced b ...

. 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 resembling a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have ...

''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, an internal binary op ...

\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-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra. Higher-dimensional categories A f ...

, where there may be non-identity

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

s between algebraic expressions, an associator is an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word is ...

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

Category theory

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

, 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 and r ...

References

* * Non-associative algebra {{algebra-stub

Ring theory

Quasigroup theory

Higher-dimensional algebra

Category theory

See also

References