Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Polynomials can be added using the associative law of addition ( grouping all their terms together into a single sum ), possibly followed by reordering, and combining of like terms.

That is, there is an identity ( no shift, everything stays where it was ), and inverses ( move everything back to where it was ), and it obeys the associative law.

A set equipped with two commutative and associative binary operations ∨ (" join ") and ∧ (" meet ") which are connected by the absorption law

For algebras that conform to the associative law, the first step is to consider the collection of all possible words formed from an alphabet.

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz ' law ; that is, the bracket is also a derivation.

When the associative law is expressed in its familiar form, one lets some symbol between two variables represent a binary operation, thus:

But if we write ƒ ( a, b ) instead of a * b then the associative law looks more like what one conventionally thinks of as a functional equation:

He obtains a partial fingerprint of Jamal Abu Zikri from traces of gunshot residue at the crime scene, but notes that it is only associative evidence and not necessarily enough evidence for a conviction in a court of law.

In 1932, Pascual Jordan attempted to axiomatize quantum theory by saying that the algebra of observables of any quantum system should be a formally real algebra which is commutative ( xy = yx ) and power-associative ( the associative law holds for products involving only x, so that powers of any element x are unambiguously defined ).

For example Lie rings and Lie algebras replace the associative law with the Jacobi identity, while Jordan rings and Jordan algebras replace the associative law with the Jordan identity.

A sufficient defining equation is the associative law:

The modular law can be seen ( and memorized ) as a restricted associative law that connects the two lattice operations similarly to the way in which the associative law λ ( μx )

In category theory a small category can be defined as a multidigraph ( with edges having their own identity ) equipped with an associative composition law and a distinguished self-loop at each vertex serving as the left and right identity for composition.

However, applying the associative law and recursive foiling allows one to expand such products.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

Premultiplied alpha has some practical advantages over normal alpha blending because premultiplied alpha blending is associative and linear interpolation gives better results, although premultiplication can cause a loss of precision and, in extreme cases, a noticeable loss of quality.

By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra.

However, during sleep, when interference is minimal, memories can be consolidated without associative interference.

One can compose functors, i. e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor G ∘ F from A to C. Composition of functors is associative where defined.

For any associative algebra A with multiplication, one can construct a Lie algebra L ( A ).

It is known that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion.

If R is commutative, then one can associate to every polynomial P in R, a polynomial function f with domain and range equal to R ( more generally one can take domain and range to be the same unital associative algebra over R ).

Red – black trees are also particularly valuable in functional programming, where they are one of the most common persistent data structures, used to construct associative arrays and sets which can retain previous versions after mutations.

Each MMU contained a 32-entry fully associative translation lookaside buffer ( TLB ) that can map 4 KB, 64 KB or 1 MB pages.

These relationships can be divided into three types: hierarchical, equivalency or associative.

Namely, Cℓ can be considered as a functor from the category of vector spaces with quadratic forms ( whose morphisms are linear maps preserving the quadratic form ) to the category of associative algebras.

Olfactory hallucinations can also appear in some cases of associative imagination, for example, while watching a romance movie, where the man gifts roses to the woman, the viewer senses the roses ' odor ( which in fact does not exist ).

it can be checked that this is again an ultrafilter, and that the operation + is associative ( but not commutative ) on and extends the addition on ; 0 serves as a neutral element for the operation + on.

If f = ∑ a < sub > n </ sub > X < sup > n </ sup > is an element of R < nowiki ></ nowiki > X < nowiki ></ nowiki >, S is a commutative associative algebra over R, I is an ideal in S such that the I-adic topology on S is complete, and x is an element of I, then we can define

The STL provides a ready-made set of common classes for C ++, such as containers and associative arrays, that can be used with any built-in type and with any user-defined type that supports some elementary operations ( such as copying and assignment ).

A set can be seen, and implemented, as a ( partial ) associative array, in which the value of each key-value pair has the unit type.

As noted in the previous section, in languages which do not directly support sets but do support associative arrays, sets can be emulated using associative arrays, by using the elements as keys, and using a dummy value as the values, which are ignored.

Similarly, we can define a weight space V < sub > λ </ sub > for any representation of a Lie group or an associative algebra.

Addition of octonions is also associative, but multiplication of octonions is non-associative.

The term is also used, especially in the description of algorithms, to mean associative array or " abstract array ", a theoretical computer science model ( an abstract data type or ADT ) intended to capture the essential properties of arrays.

For a homomorphism of unital associative R-algebras, we also demand that

He also contended that while associative learning, or memorizing ability, is equally distributed among the races, conceptual learning, or synthesizing ability, occurs with significantly greater frequency in Asians than in whites.

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space.

While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest.

Icon also includes functionality for sets and tables ( known as hashes, associative arrays, dictionaries, etc.

Built-in functions and libraries allow lists to also be used as associative arrays and ordered and unordered sets.

As We May Think also predicted many kinds of technology invented after its publication in addition to hypertext such as personal computers, the Internet, the World Wide Web, speech recognition, and online encyclopedias such as Wikipedia: " Wholly new forms of encyclopedias will appear, ready-made with a mesh of associative trails running through them, ready to be dropped into the memex and there amplified.

Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+), and also satisfies an additional property, x ⊕ x = 0 ( technically speaking, the nonnegative integers under ⊕ form an Abelian group of exponent 2 ).

This view was also thought unable to address many issues such as metaphor or associative meanings, and semantic change, where meanings within a linguistic community change over time, and qualia or subjective experience.

The SNOBOL4 variant of the language supports a number of built-in data types, such as integers and limited precision real numbers, strings, patterns, arrays, and tables ( associative arrays ), and also allows the programmer to define additional data types and new functions.

A geometry constraint engine may also be employed to manage the associative relationships between geometry, such as wireframe geometry in a sketch or components in an assembly.

As pointed out by philosophers such as Hobbes, Locke and Hume, some animals are also clearly capable of a type of " associative thinking ", even to the extent of associating causes and effects.

The comitative case ( abbreviated ), also known as the associative case ( abbreviated ), is a grammatical case that denotes companionship, and is used where English would typically use preposition " with " in the sense of " in company with " or " together with " ( other uses of " with ," e. g. with the meaning of " using ," " by means of " ( I cut bread with a knife ) would correspond to the instrumental case or related cases ).

This operation is also associative with identity, and the identity coincides with that for vertical composition.

This notion also applies to most specializations, where the multiplication must satisfy additional properties, e. g. to associative algebras or to Lie algebras.

There is also additional evidence that the neuropil may function in olfactory associative learning and memory .< ref name =" Renn ">

The associative process described by the R-W model also accounts for extinction ( see Procedures above ).

The definition of an associative K-algebra with unit is also frequently given in an alternative way.

It adds to the functionality of C ++ by also implementing design by contract, unit testing, true modules, garbage collection, first class arrays, associative arrays, dynamic arrays, array slicing, nested functions, inner classes, closures, anonymous functions, compile time function execution, lazy evaluation and has a reengineered template syntax.

Noncommutative rings and associative algebras ( rings that are also vector spaces ) are often studied via their categories of modules.