Abstract
In this paper we study the semigroup Mn(T) of all n×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 Mn(T) which contain symmetric idempotents.
Highlights
Tropical algebra is the algebra of the real numbers extended by adding an infinite negative element −∞ when equipped with the binary operations of addition and maximum
Consider the multi-machine interactive production process (MMIPP) [4] where products P1, . . . , Pm are prepared using n machines, every machine contributing to the completion of each product by producing a partial product
In 2017, we showed that a maximal subgroup of the multiplicative semigroup of n × n tropical matrices containing a nonsingular idempotent matrix E is isomorphic to the group of all invertible matrices which commute with E as groups and proved that each maximal subgroup of the multiplicative semigroup of n × n tropical matrices with the identity of the rank r is isomorphic to some maximal subgroup of the multiplicative semigroup of r × r tropical matrices with nonsingular identity
Summary
Tropical algebra ( known as max-plus algebra or maxalgebra) is the algebra of the real numbers extended by adding an infinite negative element −∞ when equipped with the binary operations of addition and maximum. The main purpose of this paper is to study the invertible matrices that commute with a nonsingular symmetric idempotent and to give a decomposition of the maximal subgroups of n × n tropical matrices containing a nonsingular symmetric idempotent. This result (see Theorem 11) develops the results obtained by Izhakian et al in [13]. In the last section, we prove that each symmetric nonsingular idempotent matrix is equivalent to a block diagonal matrix and a decomposition of the maximal subgroup containing a symmetric idempotent matrix is given (see Theorem 17)
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