The number of rectangular islands by means of distributive lattices

If A=(aij)m×nA=(aij)m×n is an m×nm×n matrix of real numbers and α,β,γ,δα,β,γ,δ are integers with 1≤α≤β≤m1≤α≤β≤m and 1≤γ≤δ≤n1≤γ≤δ≤n then the elements aijaij with α≤i≤βα≤i≤β and γ≤j≤δγ≤j≤δ form a submatrix RR which we call a rectangle   of AA. Let rr be the least element (or one of the least elements) of RR. If for every element aijaij of AA which is neighbouring to RR we have aij