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.