aXbYc

aXbYc is a symbolic notation used in abstract algebra and related fields to denote a composite operation that combines three elements a, b, and c through two binary operations X and Y. The expression is intended to illustrate how sequential binary applications can be arranged and analyzed within a single compact form.

Formally, let A be a set equipped with two binary operations X: A×A→A and Y: A×A→A. The expression a X b Y c is typically interpreted as (a X b) Y c, with parentheses indicating the order of application. Some authors may also write a X (b Y c), especially when X and Y are associative or when the intention is to emphasize different stages of combination. Because X and Y may have distinct algebraic properties, the result can depend on the chosen grouping if the operations are not associative or do not satisfy certain distributive relations.

Examples help illustrate the idea. If X is addition and Y is multiplication on the integers, with a = 2, b = 3, c = 4, then a X b Y c evaluates to (2 + 3) × 4 = 20. Conversely, if X is multiplication and Y is addition, the same numbers give (2 × 3) + 4 = 10.

In practice, aXbYc serves as a pedagogical device to discuss sequential composition of two binary operations and to explore how properties like associativity and distributivity affect simplification and evaluation. It is not a standardized notation, but can appear in didactic texts or informal discussions about composite operations. See also: binary operation, associative law, distributive law.