Maps Preserving Idempotent Relations On Matrix Spaces

Matrix algebra is an important research area of algebra. It has widely applications in other areas, for example, computer, geometry, graph theory, economics, engineering, control, etc. Preserver Problems is a very important branch of matrix algebra, there are many excellent results appear in last one hundred years. Problem of preserving idempotent relation is a young and active branch of Preserver Problems.In this thesis we first introduce background and developing state of Preserver Problems, then study automorphisms of the poset of idempotent matrices or operator algebra and problem of preserving idempotent relation. The main results obtained in this thesis are as follows:1. In Chapter 1, the origin, development andtrend, the remainder of problem preserving idempotent relation on matrix spaces or operator algebra over any field were detailed to introduce. Discuss the necessary and sufficient conditions for the existence of the problem.2. In Chapter 2, first, maps preserving idempotent relation were characterized on symmetric matrix spaces over the field whose characteritic not 2 and which has at least four elements. Then we probe the problem preserving idempotent relation over field of characteritic 2, such problem were solved in its entirety on spaces of 2 x 2 symmetric matrices3. In Chapter 3, we characterized maps preserving idempotent relation on upper-triangular matrix spaces over the field whose characteritic not 2 and which has at least four elements.4. In Chapter 4, we resolve the problem preserving idempotent relation on matrix spaces over field which has three elements.