Idempotent version of the Fréchet contingency array problem

In this paper we study the idempotent version of the so-called Fréchet correlation array problem. The problem is studied using an algebraic approach. The major result is that there exists a unique upper bound and several lower bounds. The formula for the upper bound is given. An algorithm is proposed to compute one lower bound. Another algorithm is provided to compute all lower bounds, but the number of lower bounds may be a very large number. Note that all these results are only based on the distributive lattice property of the idempotent algebraic structure.