New Types Of Generalized Inverses And Related Partial Orders

The Moore-Penrose inverse and Drazin inverse are two classical generalized inverses,which have important applications in many fields.With the development of the theory of generalized inverses,there appeared many new types of generalized inverses,such as core inverses,pseudo core inverses and(b,c)-inverses.Due to the need of statistical research,many scholars have combined generalized inverses with partial orders(pre-orders),and considered partial orders(pre-orders)based on generalized inverses.This paper mainly focuses on right core inverses,pseudo core inverses and(b,c)-inverses in rings and semigroups,as well as the orders based on core inverses and pseudo core inverses.The main contents are arranged as follows:In Chapter 2,we mainly study right core inverses in rings.Firstly,the existence criterions of right core inverses are given,and the right core invertibility of regular elements is investigated by using one-sided invertible elements.Secondly,we derive the equivalent conditions for the existence of right core inverses of elements which have universal factorization,and present the explicit expressions of right core inverses.Finally,characterizations of right core inverses of 2 x 2 matrices and companion matrices over rings are investigated.It generalizes the relevant results for core inverses.In Chapter 3,we first characterize the pseudo core invertibility of differences and products of two projections.For two projections p and q,the pseudo core invertibility of pq,p-q,p+q,pq-qp,pq+ qp and their relations are given,so that corresponding results for core inverses are extended.Furthermore,we discuss the situation when p is a projection and q is an arbitrary element in rings.Finally,we establish the relations between pseudo core invertible elements and units of rings.We show that a±(1-a(?)a)are invertible for any given pseudo core invertible element a,and give the relations between(a±(1-a(?)a))-1 and a(?),which generalizes the relevant results of Ma et al.about the core-EP inverse for complex matrices.In addition,we prove that a±(1-as(?))are invertible and give the expressions of their inverses.In Chapter 4,we focus on the(b,c)-inverse in a semigroup with involution.We firstly prove that when(ab))*=ab and(ac)*=ac,a is both left(b,c)-invertible and left(c,b)-invertible if and only if a is both(b,c)-invertible and(c,b)-invertible,which generalizes the result of Wang and Mosic in the case of b=c.Then,we investigate the equivalent conditions for ba to be(c,b)invertible and ca to be(b,c)-invertible,which extends the related results about Moore-Penrose inverses studied by Chen et al.to(b,c)-inverses.Chapter 5 is devoted to the study of the(B,C)-inverse of matrices over rings.It is well-known that when A is an invertible complex matrix,X=A-1 is the unique solution of rk(?)(A).First of all,we generalize this result to the(B,C)-inverse of matrices over rings.Then the relations between the(B,C)-inverse and the invertibility of a bordered matrix over IBN rings are also investigated.As an application,a method for calculating the(B,C)-inverse of a complex matrix is given by using this result.Finally,the expressions of onesided(B,C)-inverses studied by Benitez et al.are extended from complex matrices to matrices over rings.In Chapter 6,we consider the core partial order and core-EP pre-order in rings.Firstly,the core-subtractivity property proposed by Ferreyra and Malik is extended from complex matrices to rings,and then this property is used to characterize the core partial order.Secondly,a new characterization of core-EP order is given by using projection elements,and we improve the results of Dolinar et al.by dropping some needless conditions,then we extend the definition of core-minus partial order from complex matrices to rings.Finally,a new partial order is defined by using the core-EP order and minus partial order,and its relation with the core-minus partial order is discussed.