Abstract:
证明了(u，v)幂等矩阵与本质(m，l)幂等矩阵的互相确定关系，由此给出了求(u，v)幂等矩阵的Jordan标准形的方法，这种方法不依赖通常的求Jordan标准形的算法，只涉及到矩阵方幂的秩和u－v次单位根εi所确定的矩阵秩最后得到以矩阵秩为基本工具的，判定(u1，v1)幂等矩阵与(u2，v2)幂等矩阵相似的充分必要条件. It has been proved that (u，v)-idempotent matrices and essential (m，l)-idempotent matrices can be determined by each other. Then it gives us a method to work out the Jordan canonical form of a (u，v)-idempotent matrix，independently on the usual method of the Jordan canonical form，only referring to the ranks of matrix powers and u-v-th unity roots εi . By using ranks of matrices as a basic tool，it also obtains some sufficient and necessary conditions for a (u1，v1)-idempotent matrix to be similar to a (u2，v2)-idempotent one

Abstract:
本文研究了幂等扩张的有界分配格的同余可换性问题.利用幂等扩张的有界分配格的对偶理论,得到了同余可换的幂等扩张的有界分配格的一个充分必要条件,推广了Davey和Priestley关于有界分配格的一些结果. In this paper, we study the idempotent extended bounded distributive lattices whose congruence are permutable. By the dual theory of idempotent extended distributive lattices, we get a necessary and sufficient condition of congruence permutable idempotent extended distributive lattices. Some results obtained by Davey and Priestley on bounded distributive lattices are generalized

Abstract:
We investigate the generalized Drazin inverse of ？ over Banach spaces stemmed from the Drazin inverse of a modified matrix and present its expressions under some conditions.

Abstract:
Several basic properties of the Drazin spectrum in Banach algebras will be studied. As an application, some results on meromorphic Banach space operators will be obtained.

Abstract:
In this article, we investigate additive properties of the Drazin inverse of elements in rings and algebras over an arbitrary field. Under the weakly commutative condition of $ab = \lambda ba$, we show that $a-b$ is Drazin invertible if and only if $aa^{D}(a-b)bb^{D}$ is Drazin invertible. Next, we give explicit representations of $(a+b)^{D}$, as a function of $a, b, a^{D}$ and $b^{D}$, under the conditions $a^{3}b = ba$ and $b^{3}a = ab$.

Abstract:
A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.

Abstract:
Let and be infinite dimensional Banach spaces over the real or complex field , and let and be standard operator algebras on and , respectively. In this paper, the structures of surjective maps from onto that completely preserve involutions in both directions and that completely preserve Drazin inverse in both direction are determined, respectively. From the structures of these maps, it is shown that involutions and Drazin inverse are invariants of isomorphism in complete preserver problems. 1. Introduction In the last decades, the study of preserver problems is an active topic in operator algebra or operator space theory (see [1]). In [2], the form of involutivity-preserving maps was given by using the known results of idempotence-preserving maps, and in [3], the authors gave the characterization of additive maps preserving Drazin inverse. These results showed that involutions and Drazin inverse are invariants of isomorphism in preserver problems. Since completely positive linear maps and completely bounded linear maps are very important in operator algebra or operator space theory [4], and the concept of completely rank nonincreasing linear maps was introduced by Hadwin and Larson in [5], many mathematicians began to focus on complete preserver problems, that is, characterizations of maps on operator spaces (subsets) that preserve some property (or invariant) completely [6]. Cui and Hou discussed the completely trace-rank-preserving linear maps and the completely invertibility-preserving linear maps in [7, 8], respectively. Subsequently, in [6, 9], general surjective maps between standard operator algebras that completely preserve invertibility or spectrum and that completely preserve spectral functions are studied, respectively, where a standard operator algebra is a norm closed subalgebra of some over a Banach space containing the identity and all finite-rank operators. Recently, in [10], the authors discussed completely idempotents preserving surjective maps and completely square-zero operators preserving surjective maps. These results showed that idempotents and square-zero operators are invariants of isomorphism in complete preserver problems. Since involutions and Drazin inverse are closely related to idempotents, it is interesting to consider whether the involutions and Drazin inverse are still invariants of isomorphism in complete preserver problems. Let and be Banach spaces over the real or complex field , and let be the Banach algebra of all bounded linear operators from to . An operator is called an involution (idempotent) if ( ), denoted by