The tropical matrix groups with symmetric idempotents (Q1727121)

Summary: In this paper we study the semigroup \(M_n(\mathbb{T})\) of all \(n \times n\) tropical matrices under multiplication. We give a description of the tropical matrix groups containing a diagonal block idempotent matrix in which the main diagonal blocks are real matrices and other blocks are zero matrices. We show that each nonsingular symmetric idempotent matrix is equivalent to this type of block diagonal matrix. Based upon this result, we give some decompositions of the maximal subgroups of \(M_n(\mathbb{T})\) which contain symmetric idempotents.