Symmetric Properties of (<i>b</i>,<i>c</i>)-Inverses

Symmetric Properties of (<i>b</i>,<i>c</i>)-Inverses

Let <i>b</i> and <i>c</i> be two elements in a semigroup <i>S</i>. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo...

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Main Authors: Guiqi Shi, Jianlong Chen
Format: Article
Language:English
Published: MDPI AG 2022-08-01
Series:Mathematics
Subjects:
generalized inverse
<i>(b,c)</i>-inverse
inner <i>(b,c)</i>-inverse
outer inverse
Online Access:https://www.mdpi.com/2227-7390/10/16/2948
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author Guiqi Shi
Jianlong Chen
author_facet Guiqi Shi
Jianlong Chen
author_sort Guiqi Shi
collection DOAJ
description Let <i>b</i> and <i>c</i> be two elements in a semigroup <i>S</i>. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-inverse is an important outer inverse because it unifies many common generalized inverses. This paper is devoted to presenting some symmetric properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-inverses and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>-inverses. We first find that <i>S</i> contains a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible element if and only if it contains a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible element. Then, for four given elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></mrow></semantics></math></inline-formula> in <i>S</i>, we prove that <i>a</i> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible and <i>d</i> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>b</mi><mi>d</mi></mrow></semantics></math></inline-formula> is invertible along <i>c</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mi>c</mi><mi>a</mi></mrow></semantics></math></inline-formula> is invertible along <i>b</i>. Inspired by this result, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertibility is characterized by one-sided invertible elements. Furthermore, we show that <i>a</i> is inner <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible and <i>d</i> is inner <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible if and only if <i>c</i> is inner <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible and <i>b</i> is inner <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>d</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible.
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spelling doaj.art-b61f4587763e484a928ff0633a7639672023-12-01T23:57:47ZengMDPI AGMathematics2227-73902022-08-011016294810.3390/math10162948Symmetric Properties of (<i>b</i>,<i>c</i>)-InversesGuiqi Shi0Jianlong Chen1School of Mathematics, Southeast University, Nanjing 210096, ChinaSchool of Mathematics, Southeast University, Nanjing 210096, ChinaLet <i>b</i> and <i>c</i> be two elements in a semigroup <i>S</i>. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-inverse is an important outer inverse because it unifies many common generalized inverses. This paper is devoted to presenting some symmetric properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-inverses and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>-inverses. We first find that <i>S</i> contains a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible element if and only if it contains a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible element. Then, for four given elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></mrow></semantics></math></inline-formula> in <i>S</i>, we prove that <i>a</i> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible and <i>d</i> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>b</mi><mi>d</mi></mrow></semantics></math></inline-formula> is invertible along <i>c</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mi>c</mi><mi>a</mi></mrow></semantics></math></inline-formula> is invertible along <i>b</i>. Inspired by this result, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertibility is characterized by one-sided invertible elements. Furthermore, we show that <i>a</i> is inner <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible and <i>d</i> is inner <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible if and only if <i>c</i> is inner <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible and <i>b</i> is inner <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>d</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula>-invertible.https://www.mdpi.com/2227-7390/10/16/2948generalized inverse<i>(b,c)</i>-inverseinner <i>(b,c)</i>-inverseouter inverse
spellingShingle Guiqi Shi
Jianlong Chen
Symmetric Properties of (<i>b</i>,<i>c</i>)-Inverses
Mathematics
generalized inverse
<i>(b,c)</i>-inverse
inner <i>(b,c)</i>-inverse
outer inverse
title Symmetric Properties of (<i>b</i>,<i>c</i>)-Inverses
title_full Symmetric Properties of (<i>b</i>,<i>c</i>)-Inverses
title_fullStr Symmetric Properties of (<i>b</i>,<i>c</i>)-Inverses
title_full_unstemmed Symmetric Properties of (<i>b</i>,<i>c</i>)-Inverses
title_short Symmetric Properties of (<i>b</i>,<i>c</i>)-Inverses
title_sort symmetric properties of i b i i c i inverses
topic generalized inverse
<i>(b,c)</i>-inverse
inner <i>(b,c)</i>-inverse
outer inverse
url https://www.mdpi.com/2227-7390/10/16/2948
work_keys_str_mv AT guiqishi symmetricpropertiesofibiiciinverses
AT jianlongchen symmetricpropertiesofibiiciinverses

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