All examples in this section involve associative operators, thus we shall use the terms left / right inverse for the unital magma-based definition, and quasi-inverse for its more general version.

The original German version uses a rule system in which the Gamemaster and players draw associative, tarot-like cards instead of rolling dice to determine the outcome of an event ( this system is called the Arcana system in the original German version ).

Since this holds true when performing addition on any real numbers, we say that " addition of real numbers is an associative operation.

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

* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

However, this operation may not be appropriate for all applications, since it is not associative.

* If the operation is associative, ( ab ) c = a ( bc ), then the value depends only on the tuple ( a, b, c ).

* Lookup table, a data structure, usually an array or associative array, often used to replace a runtime computation with a simpler array indexing operation

The Lie bracket is not an associative operation in general, meaning that need not equal.

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

The operation is required to be associative so that for all x, y and z, but need not be commutative so that does not have to equal ( contrast to the standard multiplication operator on real numbers, where ).

Semigroups must not be confused with quasigroups which are sets with a not necessarily associative binary operation such that division is always possible.

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

If the operation is associative then if an element has both a left inverse and a right inverse, they are equal.

An example is the associative axiom for a binary operation, which is given by the equation x * ( y * z ) = ( x * y ) * z.

Briefly, a ring is an abelian group with an additional binary operation that is associative and is distributive over the abelian group operation.

) Because of the associative nature of the " and " operation in propositional logic, this becomes a functional equation saying that the function g such that

: is an associative binary operation.

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.

All of the preceding categories have the identity map as identity arrow and composition as the associative operation on arrows.

Any monoid ( any algebraic structure with a single associative binary operation and an identity element ) forms a small category with a single object x.

* An internal operation ( addition ) which is associative, commutative, distributive and with zero and unity elements

* Associativity: composition of morphisms is always associative.

In mathematics, the associative property is a property of some binary operations.

Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.

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:

The two methods produce the same result ; string concatenation is associative ( but not commutative ).

* Addition and multiplication of complex numbers and quaternions is associative.

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

In short, composition of maps is always associative.

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.

The second assumption is that the operator must respect the associative rule:

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.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

An associative R-algebra is an additive abelian group A which has the structure of both a ring and an R-module in such a way that ring multiplication is R-bilinear:

One of the main starting points of the Alain Connes ' direction in noncommutative geometry is his spectacular discovery ( and independently by Boris Tsygan ) of a very important new homology theory associated to noncommutative associative algebras and noncommutative operator algebras, namely the cyclic homology and its relations to the algebraic K-theory ( primarily via Connes-Chern character map ).

It also harbours a very dynamic associative life, with its numerous sports and cultural sections, as well as a number of other autonomous organizations.

With a long history of struggle against unpopular state policies, forming notable politicians and intellectuals along the way, it also harbours a very dynamic associative life.

Further questions occur about navigation, associative information access, programming and communication within very large data sets.

They receive abundant afferrences from the striatum ( mainly from the associative striatum ) with the same very peculiar synaptology as the pallidum.