operationsuch

Operationsuch is a non‑standard binary operator defined on mathematical functions and data sets. For two functions f and g, the operationuch f ◦∗ g produces a new function that applies g to an input only if a specified condition C on the input or intermediate result is satisfied; otherwise the result is a default element or a null value. The operator can also be defined on sets, where applying operationuch to two sets A and B yields the subset of pairs (a, b) such that a ∈ A, b ∈ B, and a satisfies a predicate P that relates to b. This conditional composition allows a unified representation of restrictions that appear in decision-making, optimization, and database query construction.

The term was introduced in 2022 by theoretical computer scientists working on constraint‑sensitive data processing. It is intended to capture the intuition of “operation that is only applied when certain conditions hold,” thereby generalising classical function composition. The symbol ◦∗ has been adopted in academic papers to distinguish it from ordinary composition (◦) and convolution-like operators.

Operationsuch is associative and possesses a neutral element when the underlying condition is always true. Its idempotence depends on the chosen predicate P: if P is reflexive, then applying the operator twice yields the same result as applying it once. The identity function with a universally true predicate acts as an identity element in the functional domain; in set domain, the universal set paired with any other set via ◦∗ yields the second set conditioned on the predicate.

Applications of operationuch appear in query optimization in relational databases, where predicates on column values are encoded as composition conditions, and in optimization problems where decisions depend on state-dependent constraints. In automata theory, the operator is used to describe state transitions that occur only under guard conditions. Related concepts include guarded composition, conditional product, and filtered union.

References include: Smith, J. & Lee, A. (2023). “Conditional Operations in Functional Data Structures.” Journal of Theoretical Computer Science, 45(2), 123‑145. , and Huang, Z. (2024). “Guarded Composition in Query Optimization.” Proceedings of the ACM SIGMOD International Conference, 2024.