Sign-nonsingular matrices and matrices with unbalanced determinant in symmetrised semirings (Q1970518)

Let \(\overline{\mathbb{R}}= \mathbb{R}\cup\{- \infty\}\) be the set of reals extended by \(-\infty\), and let \(S= \overline{\mathbb{R}}\times \overline{\mathbb{R}}\). The semiring \((\overline{\mathbb{R}},\oplus, \otimes)\), where \(a\oplus b= \max\{a, b\}\), \(a\otimes b= a+b\), can be identified by using the morphism \(t\mapsto (t,-\infty)\) with a subset of the symmetrised semiring \((S,\oplus,\otimes)\), where \((a,a')\oplus (b,b')= (a\oplus b,a'\oplus b')\), \((a,a')\otimes (b,b')= (a\otimes b\oplus a'\otimes b',a\otimes b'\oplus a'\otimes b)\). We say that \(x= (x',x'')\) balances \(y= (y',y'')\) and denote \(x\nabla y\) if \(x'\oplus y''= x''\oplus y'\). An element \(x= (a,b)\) is called sign-positive (resp. sign-negative) if \(a>b\) (resp. \(a<b\)) or \(x= (-\infty,-\infty)\). The determinant of an \(n\times n\) matrix \(A\) over \(S\) is defined as \[ \text{det}(A)= \sum^\oplus_\sigma \text{sgn}(\sigma) \prod^\otimes_{i\in\mathbb{N}} a_{i,\sigma(i)}, \] where \(\text{sgn}(\sigma)= 0\) if \(\sigma\) is even and \(\text{sgn}(\sigma)= (-\infty, 0)\) if \(\sigma\) is odd. The matrix \(A\) is said to have balanced determinant if \(\text{det}(A)\nabla(- \infty,-\infty)\), otherwise it is said to have unbalanced determinant. An \(n\times n\) matrix over the set \(\{0,1,-1\}\) is said to be sign-nonsingular if at least one term of its standard determinant expansion is nonzero and all nonzero terms have the same sign. The main result concerns a relationship existing between sign-nonsingular \((0,1,-1)\) matrices and matrices with unbalanced determinant.