Associativity

Associativity is the law that regrouping repeated applications of an operation leaves the result unchanged. For a binary operation, it means (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c).

Binary operation: A rule that combines two elements of a set to produce another element. Associativity is a condition imposed on binary operations.
Matrix multiplication: The operation that multiplies matrices by combining rows of one matrix with columns of another. Matrix products are associative, so products can be regrouped without changing their value.
Commutativity: The law that swapping the inputs of an operation leaves its result unchanged. Associativity changes grouping; commutativity changes the order of inputs.
Semigroup: A set with an associative binary operation. Associativity is the defining law of a semigroup’s operation.
Set: A collection of distinct objects considered as a whole. An operation’s elements are drawn from a specified set.
Function composition: The operation of applying one function after another. Composed functions associate, allowing chains of functions to be grouped either way.
Distributivity: A law relating two operations, in which one operation distributes across another. Distributivity links operations rather than merely regrouping one operation.
Monoid: A semigroup with an identity element. Adding an identity to an associative operation produces a monoid.
Function: A rule assigning each input exactly one output. A binary operation is a function of two inputs.
Concatenation: The operation of joining sequences end to end. Sequence concatenation is associative, enabling regrouping of strings or lists.

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