Group Structure and Narrowing Effectiveness of Best-Possible Bounds for Copulas Specified at a Single Interior Point

We express the group structure of sets of a copula , survival copula , the dual of a copula , and the co-copula , give the best-possible bounds for the last three functions where the value of every function is known at a single point. Especially, we find an general narrowing effectiveness coefficient for , compare the values of where to thoses for being other values in .