Geodesic
On elements whose (b, c)-inverse is idempotent in a monoid
Filomat, Tome 36 (2022) no. 14, p. 4645

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In this paper, we investigate the elements whose (b, c)-inverse is idempotent in a monoid. Let S be a monoid and a, b, c ∈ S. Firstly, we give several characterizations for the idempotency of a ||(b,c) as follows: a ||(b,c) exists and is idempotent if and only if cab = cb, cS = cbS, Sb = Scb if and only if both a ||(b,c) and 1 ||(b,c) exist and a ||(b,c) = 1 ||(b,c) , which establish the relationship between a ||(b,c) and 1 ||(b,c). They imply that a ||(b,c) merely depends on b, c but is independent of a when a ||(b,c) exists and is idempotent. Particularly, when b = c, more characterizations which ensure the idempotency of a ||b by inner and outer inverses are given. Finally, the relationship between a ||b and a ||b n for any n ∈ N + is revealed.
DOI : 10.2298/FIL2214645Z
Classification : 15A09, 20M99
Keywords: (b, c)-inverse, inverse along an element, group inverse, idempotent 
Haiyang Zhu; Jianlong Chen; Yukun Zhou. On elements whose (b, c)-inverse is idempotent in a monoid. Filomat, Tome 36 (2022) no. 14, p. 4645 . doi: 10.2298/FIL2214645Z
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     title = {On elements whose (b, c)-inverse is idempotent in a monoid},
     journal = {Filomat},
     pages = {4645 },
     year = {2022},
     volume = {36},
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     doi = {10.2298/FIL2214645Z},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2214645Z/}
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