Semigroup Of Upper Triangular Matrix

This paper mainly discusses the properties of the upper triangular matrix semi-group.Using S denotes the set of all upper triangular matrixes,it is clear that S composes a semigroup according to the multiplication of the matrix.In this paper,we study the regular elements,idempotent and maximal regular subgroups of upper triangular matrix semigroups and Green Relations.In the first chapter,we give the relation between the regular element and the matrix rank in the upper triangular ma-trix semigroup S,and prove that the sufficient and necessary condition of the upper triangular matrix A to be regular in S is rank(A)= n-r.In the second chapter,I con-struct a class of maximal regular subgroups.In the third chapter,we give the sufficient and necessary condition for the upper triangular matrix A to be an idempotent in S,which is the reversible upper triangular matrix P of S such that PAP-1 is a diagonal matrix,and the diagonal elements are only 0 and 1.In chapter 4,the Green R.relation and the Green L relation of the upper triangular matrix semigroup are described the linear correlation of the row vector and the column vector.In the last chapter of this paper,we propose a subgroup of special triangular matrix semigroups for matrix addition.In this paper,we discuss some of the most typical problems in semigroups by five chapters,and give a set of methods and ideas for studying semigroups by means of triangular matrix semigroups.I hope to give readers some new ideas.