A note on minimal matrix representation of closure operations

A matrixM withn columns represents a closure operationF(A), (A⊂X, |X|=n) if for anyA, any two rows equal in the columns corresponding toA are also equal inF(A). Letm(F) be the minimum number of rows of the matrices representingF. Lower and upper estimates on maxm(F) are given where max runs over the set of all closure operations onn elements.