Tokomplement

Tokomplement is a mathematical operation defined on a universal set U together with a distinguished token subset T ⊆ U. For any A ⊆ U, the tokomplement of A with respect to T, denoted A^tok, is defined as T \ A. This construction generalizes the usual set complement by restricting attention to elements that carry a token, effectively masking all elements outside T.

Basic properties of tokomplement arise from the restriction to T. The set A^tok is always a subset of T, and A^tok is empty precisely when A contains T. If T equals U, tokomplement reduces to the classical complement U \ A. Double application satisfies (A^tok)^tok = A ∩ T. The operation interacts with union and intersection in a way that reflects the restriction: (A ∪ B)^tok = A^tok ∩ B^tok, while (A ∩ B)^tok = A^tok ∪ B^tok. These relations illustrate how restricting to a token substructure alters standard Boolean laws.

Examples help illustrate the behavior. Take U = {a, b, c, d} with T = {a, c}. For A = {a, b}, A^tok = {c}. For B = {c, d}, B^tok = {a}. Then A^tok ∪ B^tok = {a, c} while (A ∩ B)^tok = {a, c}, demonstrating the reflected duality under restriction.

Context and usage: The term tokomplement appears in hypothetical and speculative mathematical discussions as a device to illustrate how classical set operations change when limited to a token-marked substructure. It is used mainly as a teaching aid for exploring dualities and the effects of restricting operations to a subset. The concept is not tied to a widely adopted formal theory and is primarily of theoretical or illustrative interest.