Some Studies Of Tropical Matrices

In this thesis we study the semigroup of tropical matrices under multi-plication.Firstly,We give the normal form of an idempotent tropical matrix and prove that every maximal subgroup of n×n tropical matrices containing E is equal to{EM|M∈GL_n(T),M E=EM},where E is a nonsingular idempotent tropical matrix and GL_n(T)denotes the set of all n×n monomial matrices.On this basis,we give the descriptions of the maximal subgroups containing an n×n idempotent tropical matrix of rank r.We also prove that there is a natural and canonical embedding of such maximal subgroups into the group of units of the semigroup of r×r tropical matrices.As a consequence of our main result,we directly give a faithful representation of a subgroup of the semigroup of all n×n tropical matrices into the group GL_n(T).Furthermore,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 of all n×n tropical matrices which contain symmetric idempotents.Secondly,we introduce and study two actions of the invertible matrices group on the set of all n×n tropical matrices.The connections be-tween the equivalence classes determined by Green’s relations R and L and the orbits determined by the two actions are discussed.Thirdly,we study the regular D-classes of the semigroup of all n×n tropical matrices under multiplication and give a partition of a nonsingular regular D-class.Fourthly,we give the characterization of matrices having generalized inverses.Also,we introduce and study a space decomposition of a matrix,and prove that a matrix is space decomposable if and only if it has a generalized inverse.We establish necessary and sufficient conditions for a matrix to possess various types of g-inverses.Finally,we study the subgroup structure of the semigroup of all n×n Boolean matrices under multiplication and we prove that every subgroup of all n×n Boolean matrices is embedded in the symmetric group S_n.