Polyassociativity

Polyassociativity is a property of polyadic (n-ary) operations, where an operation takes n inputs for a fixed n ≥ 2. An n-ary operation f on a set A is said to be polyassociative if evaluating a sequence of inputs with repeated applications of f yields the same result regardless of how the applications are nested or bracketed. In other words, any two fully parenthesized expressions built from a list of inputs that are combined using blocks of n arguments evaluate to the same element.

In universal algebra, structures with an associative n-ary operation are called polyadic algebras or polyadic semigroups. The polyassociativity condition generalizes the familiar binary associativity: when n = 2, polyassociativity reduces to ordinary associativity of a binary operation.

Examples help illustrate the concept. If A is a commutative monoid with the usual addition, the extended n-ary sum t(a1,..., an) = a1 + ... + an is polyassociative, since addition is associative and the grouping of terms does not affect the total. Similarly, the n-ary max operation t(a1,..., an) = max(a1,..., an) is polyassociative because taking the maximum is independent of how the inputs are grouped.

Not every n-ary operation is polyassociative; it is a nontrivial property that may require specific algebraic conditions to hold. Polyassociativity is a central notion in the study of polyadic or n-ary algebraic systems and informs how complex expressions built from n-ary operations can be reliably evaluated.

See also: associative law, n-ary operation, polyadic algebra, polyadic semigroup.