On square roots of matrices over the max-time semiring ℝ+

Abstract

Abstract The set of all nonnegative real numbers ℝ+ with respect to two operations ⊕ : (a, b) → max{a, b} and ⊙ : (a, b) → ab forms a semiring that is called a max-times semiring. A square matrix A over the semiring has a max-times square root if there is a matrix B over the semiring such that B ⊙ B = A. Whereas, a square matrix A over the semiring ℝ+ with respect to the tow usualy addition and multipliacation operation has a conventional square root if there is a matrix B over the semiring such that B × B = A. We begin this study by giving an example of an 3×3 upper triangular matrix over ℝ+ that has both conventional and max-times square roots. In this paper, we do research on necessary and sufficient conditions of matrix having max-times square roots and compare it to the conventional one. Furthermore, we determine necessary and sufficient conditions of such matrix having either one, both, or neither one. Finally, we investigate the numbers of square roots, if any, in each case under all conditions.