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Symmetric Properties of ( b , c )-Inverses

Author

Listed:
  • Guiqi Shi

    (School of Mathematics, Southeast University, Nanjing 210096, China)

  • Jianlong Chen

    (School of Mathematics, Southeast University, Nanjing 210096, China)

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.

Suggested Citation

  • Guiqi Shi & Jianlong Chen, 2022. "Symmetric Properties of ( b , c )-Inverses," Mathematics, MDPI, vol. 10(16), pages 1-12, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:16:p:2948-:d:889129
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