Generators of matrix incidence algebras

Let n E Z+ and let K be a field. Let ~ be a partial order on {1, 2, ... , n}. Let An(:::;) be the matrix incidence algebra consisting of those n x n matrices A = (ai,j) with entries in K, satisfying ai,j 0 whenever i 1:. j. For a subset £ ~ An (::5), a necessary and sufficient condition that the algebra generated by £ u {I} is An(::5) is that (i) for every 1 :::; i, j :::; n with i =1= j, there exists A E £ such that ai,i =1= aj,j and (ii) for every i ~ j with j covering i, there exists B E span £ such that bi,j =1= 0 and bi,i = bj,j. If the characteristic of K is zero or > n, the algebra An (=) is singly generated and, if ::5 is not equality, An C::;) has two generators.