Symmetric Properties of ( b, c )-Inverses

Guiqi Shi and Jianlong Chen ()
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Guiqi Shi: School of Mathematics, Southeast University, Nanjing 210096, China
Jianlong Chen: School of Mathematics, Southeast University, Nanjing 210096, China

Mathematics, 2022, vol. 10, issue 16, 1-12

Abstract: Let b and c be two elements in a semigroup S . The ( b , c ) -inverse is an important outer inverse because it unifies many common generalized inverses. This paper is devoted to presenting some symmetric properties of ( b , c ) -inverses and ( c , b ) -inverses. We first find that S contains a ( b , c ) -invertible element if and only if it contains a ( c , b ) -invertible element. Then, for four given elements a , b , c , d in S , we prove that a is ( b , c ) -invertible and d is ( c , b ) -invertible if and only if a b d is invertible along c and d c a is invertible along b . Inspired by this result, the ( b , c ) -invertibility is characterized by one-sided invertible elements. Furthermore, we show that a is inner ( b , c ) -invertible and d is inner ( c , b ) -invertible if and only if c is inner ( a , d ) -invertible and b is inner ( d , a ) -invertible.

Keywords: generalized inverse; (b,c) -inverse; inner (b,c) -inverse; outer inverse (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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