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In this paper we study the positive resolvent values of positive operators respectively of positive elements in Banach lattice ordered algebras. In the matrix case these values are just the inverse M-matrices. One of the main results is the following: Let A be a Banach lattice ordered algebra. A positive invertible element x ∈ A is a resolvent value of a positive element y ∈ A if and only if the element x satisfies the negative principle: If a ∈ A, λ < 0 and xa ≤ λa then xa ≤ 0.