Left and right generalized inverses
Left and right generalized inverses
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DOI：
10.1016/j.laa.2016.08.010
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发表时间：
2016-12
影响因子：
1.1
通讯作者：
M. Drazin
中科院分区：
文献类型：
--
作者：
M. Drazin
本文研究了一种定义大类“(b, c)-逆”的左、右版本的方法，该类由作者在 (2012)[6] 中引入：给定任何半群 S 和任何 a, b, c ∈ S，则 a 被称为左 (b, c) 可逆，如果 b ∈ S c a b，并且 x ∈ S 被称为 a 的左 (b, c) 逆，如果 x ∈ S c 且 x a b= b，并且对偶 c ∈ c a b S, z∈ S b 且 c a z= z 表示 a 的右 (b, c) 逆 z。结果表明，a 的左和右 (b, c)-可逆性一起意味着 (b, c)-可逆性，在这种情况下，a 的每个左 (b, c)-逆元也是右 (b, c)-逆元，反之亦然，则 a 的所有左或右 (b, c)-逆元都重合。当b= c（例如对于摩尔-彭罗斯逆或作者的伪逆）时，左(b, b)-可逆性与每个强π-正则半群中的右(b, b)-可逆性一致。 Vaserstein 和 Goodearl 的一个基本结果保证了稳定范围 1 的 Bass 性质的左右对称性，它从两侧逆元扩展到左或右逆元，并且对于中心 b，扩展到左或右 (b, b)-逆元。
This article examines a way to define left and right versions of the large class of “(b, c)-inverses” introduced by the writer in (2012)[6]: Given any semigroup S and any a, b, c∈ S, then a is called left (b, c)-invertible if b∈ S c a b, and x∈ S is called a left (b, c)-inverse of a if x∈ S c and x a b= b, and dually c∈ c a b S, z∈ S b and c a z= z for right (b, c)-inverses z of a. It is shown that left and right (b, c)-invertibility of a together imply (b, c)-invertibility, in which case every left (b, c)-inverse of a is also a right (b, c)-inverse, and conversely, and then all left or right (b, c)-inverses of a coincide. When b= c (eg for the Moore-Penrose inverse or for the pseudo-inverse of the author) left (b, b)-invertibility coincides with right (b, b)-invertibility in every strongly π-regular semigroup. A fundamental result of Vaserstein and Goodearl, which guarantees the left-right symmetry of Bass's property of stable range 1, is extended from two-sided inverses to left or right inverses, and, for central b, to left or right (b, b)-inverses.