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The tropical matrix groups with symmetric idempotents. (English) Zbl 1417.15038

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.

MSC:

15A80 Max-plus and related algebras
20M10 General structure theory for semigroups
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