The number of rectangular islands by means of distributive lattices

If A = ( a i j ) m × n is an m × n matrix of real numbers and α , β , γ , δ are integers with 1 ≤ α ≤ β ≤ m and 1 ≤ γ ≤ δ ≤ n then the elements a i j with α ≤ i ≤ β and γ ≤ j ≤ δ form a submatrix R which we call a rectangle of A . Let r be the least element (or one of the least elements) of R . If for every element a i j of A which is neighbouring to R we have a i j < r then R is called a rectangular island of A . More precisely, R is called a rectangular island if whenever ( i , j ) ∈ ( { 1 , … , m } × { 1 , … , n } ) ∖ ( { α , … , β } × { γ , … , δ } ) , ( k , ℓ ) ∈ { α , … , β } × { γ , … , δ } , | i − k | ≤ 1 and | j − ℓ | ≤ 1 then a i j < r . rst aim of the present paper is to determine the maximum of the number of rectangular islands of m × n matrices, for any fixed pair ( m , n ) of positive integers. The second aim is to point out that a purely lattice theoretic result on weak bases of distributive lattices in [G. Czédli, A.P. Huhn, E.T. Schmidt, Weakly independent subsets in lattices, Algebra Universalis 20 (1985) 194–196] is useful in combinatorics.