Characterization of idempotent 2-copulas

A 2-copula A induces a transition probability function p(A) via p(A)(x, S) = d/dx integral(s) partial derivative/partial derivative tA(x, t) dt. where S is an element of B, B denoting the Lebesgue measurable subsets of [0, 1]. We say that a set S is invariant under A if p(A)(x, S) =chi(S)(x) for almost all x is an element of [0, 1], chi(S) being the characteristic function of S. The sets S invariant under A form a sub-sigma-algebra of the Lebesgue measurable sets, which we denote B-A. A set S is an element of B-A is called an atom if it has positive measure and if for any S' is an element of B-A, lambda(S' boolean AND S) is either lambda(S) or 0. A 2-copula F is idempotent if F * F = F. Here * denotes the product defined in [1]. Idempotent 2-copulas are classified and characterized as follows: (i) An idempotent F is said to be nonatomic if B-F contains no atoms. If F is a nonatomic idempotent, then it is the product of a left invertible copula and its transpose. That is, there exists a copula B such that B * B-T = F, and B-T * B = M, where M(x, y) = min(x, y). (ii) An idempotent F is said to be totally atomic if there exist essentially disjoint atoms S-n is an element of B-F with Sigma(n) lambda(S-n) = 1. If F is a totally atomic idempotent, then it is conjugate to an ordinal sum of copies of the product copula. That is, there exists a copula C satisfying C * C-T = C-T * C = M and a partition P of [0, 1] such that F = C * (circle plus F-P(k)) * C-T (1) where each component F-k in the ordinal sum is the product copula P. (iii) An idempotent F is said to be atomic (but not totally atomic) if B-F contains atoms but the sum of the measures of a maximal collection of essentially disjoint atoms is strictly less than 1. In this mixed case, there exists a copula C invertible with respect to M and a partition P of [0, 1] for which (1) holds, with F-1 being a nonatomic idempotent copula and with F-k = P for k > 1. Some of the immediate consequences of this characterization are discussed.