AdditivAddition

AdditivAddition is a theoretical binary operation defined on a set with a baseline addition. For elements a and b in a set S, it is written as a ⊕ b = a + b + f(a,b), where f: S×S → S is an auxiliary function that encodes extra additive contributions such as synergy, overlap, or context bias. If f is identically zero, ⊕ coincides with ordinary addition.

Properties of AdditivAddition depend on the choice of f. If f is symmetric (f(a,b) = f(b,a)), the operation ⊕ is commutative. The existence and location of an identity element depend on f; in general, the identity, if it exists, differs from the baseline identity. Associativity imposes a condition on f that must be verified for each case.

Concrete instantiations help illustrate the idea. Example 1: with S = nonnegative integers, baseline + and f(a,b) = k ≥ 0, a ⊕ b = a + b + k is commutative and associative (the identity is −k). Example 2: f(a,b) = gcd(a,b) yields a ⊕ b = a + b + gcd(a,b), which is commutative but not automatically associative.

Applications include modeling synergistic effects in economics, correcting for overlap in data aggregation, and designing aggregators for multi‑criteria decision making. The concept provides a way to study how additive contributions may be augmented by context while preserving the general additive framework.