Completely positive matrices over Boolean algebras and their CP-rank

Abstract

Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [n2/4]. 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.

Keywords

• Boolean matrices
• matrices over semirings
• completely positive matrices
• diagonally dominant matrices
• semiring homomorphisms
• 15B33
• 15B34
• 15B48
• 16A78