To cite this article: Preeti Mohindru (2015) The Drew-Johnson-Loewy conjecture for matrices over max-min semirings, Linear and Multilinear Algebra, 63:5, 914-926, Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive (CP) real matrix is at most n 2 /4 . In this paper, we prove this conjecture for n × n CP matrices over max-min semirings.

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Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [︀ n 2 /4 ]︀ . In this paper, we prove this conjecture for n × n completely positive matrices over Boolean algebras (finite or infinite). In addition, we formulate various CP-rank inequalities of completely positive matrices over special semirings using semiring homomorphisms. Definition 1.4. [11] A semiring is a set S together with two operations ⊕ and ⊗ and two distinguished elements0, 1 in S with 0 ≠ 1, such that 1. (S, ⊕, 0) is a commutative monoid, 2. (S, ⊗, 1) is a monoid, 3. ⊗ is both left and right distributive over ⊕, 4. The additive identity 0 satisfies the property r ⊗ 0 = 0 ⊗ r = 0, for all r ∈ S.

We use the ϵ-determinant introduced by Ya-Jia Tan to define a family of ranks of matrices over certain semirings. We show that these ranks generalize some known rank functions over semirings such as the determinantal rank. We also show that this family of ranks satisfies the rank-sum and Sylvester inequalities. We classify all bijective linear maps which preserve these ranks.

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