Abstract
We present equivalent conditions of reverse order law for the (b, c)-inverse \({(a_1a_2)^{(b, c)}=a_2^{(b, s)}a_1^{(t, c)}}\) to hold in a semigroup. Also, we study various mixed-type reverse order laws for the (b, c)-inverse. As a consequence, we get results related to the reverse order law for the inverse along an element. More general case of reverse order law, precisely the rule \({(a_1a_2)^{(b_3, c_3)}=a_2^{(b_2,c_2)}a_1^{(b_1, c_1)}}\) is considered too.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Benitez and E. Boasso, The inverse along an element in rings, arXiv:1507.05410 (2015).
J. Benitez and E. Boasso, The inverse along an element in rings with an involution, Banach algebras and C*-algebras, Linear Multilinear Algebra, Doi:10.1080/03081087.2016.1183559.
Djordjević D.S.: Unified approach to the reverse order rule for generalized inverses. Acta Sci. Math. (Szeged) 67, 761–776 (2001)
Djordjević D.S., Dinčić N.C.: Reverse order law for the Moore–Penrose inverse. J. Math. Anal. Appl. 361, 252–261 (2010)
D.S. Djordjević and V. Rakočević, Lectures on Generalized Inverses, University of Niš, Faculty of Sciences and Mathematics (Niš 2008).
Djordjević D.S., Wei Y.M.: Outer generalized inverses in rings. Comm. Algebra 33, 3051–3060 (2005)
Drazin M.P.: Pseudo-inverses in associative rings and semigroups. Amer. Math. Monthly 65, 506–514 (1958)
Drazin M.P.: A class of outer generalized inverses. Linear Algebra Appl. 436, 1909–1923 (2012)
Drazin M.P.: Commuting properties of generalized inverses. Linear Multilinear Algebra 61, 1675–1681 (2013)
Drazin M.P.: Generalized inverses: Uniqueness proofs and three new classes. Linear Algebra Appl. 449, 402–416 (2014)
Greville T.N.E.: Note on the generalized inverse of a matrix product. SIAM Rev. 8, 518–521 (1966)
Hartwig R.E.: The reverse order law revisited. Linear Algebra Appl. 76, 241–246 (1986)
Hartwig R.E.: Block generalized inverses. Arch. Ration. Mech. Anal. 61, 197–251 (1976)
Kantún-Montiel G.: Outer generalized inverses with prescribed ideals. Linear Multilinear Algebra 62, 1187–1196 (2014)
Ke Y.Y., Chen J.L.: The Bott–Duffin (e, f)-inverses and their applications. Linear Algebra Appl. 489, 61–74 (2016)
Y. Y. Ke, Z. Wang and J. L. Chen, The (b, c)-inverse for products and lower triangular matrices (submitted).
Y. Y. Ke, D. S. Cvetković-Ilić, J. L. Chen and J. Višnjić, New results for (b, c)-inverses (submitted).
Mary X.: On generalized inverses and Green’s relations. Linear Algebra Appl. 434, 1836–1844 (2011)
Mary X., Patrício P.: The inverse along a lower triangular matrix. Appl. Math. Comput. 219, 886–891 (2012)
D. Mosić, Characterizations of the image-kernel (p, q)-inverses, Bull. Malays. Math. Sci. Soc. 2, (2015), doi:10.1007/s40840-015-0242-x.
Mosić D., Djordjević D.S.: Further results on the reverse order law for the Moore–Penrose inverse in rings with involution. Appl. Math. Comput. 218, 1478–1483 (2011)
Mosić D., Djordjević D.S., Kantún-Montiel G.: Image-kernel (P, Q)-inverses in rings. Electron. J. Linear Algebra. 27, 272–283 (2014)
Mosić D., Djordjević D.S.: Inner image-kernel (p, q)-inverses in rings. Applied Math. Comput. 239, 144–152 (2014)
L. Wang, J. L. Chen and N. Castro-González, Characterizations of the (b, c)-inverse in a ring, arXiv:1507.01446 (2015).
Zhu H.H., Chen J.L., Patrício P.: Further results on the inverse along an element in semigroups and rings. Linear Multilinear Algebra 64, 393–403 (2016)
H. H. Zhu, J. L. Chen and P. Patrício, Reverse order law for the inverse along an element, Linear Multilinear Algebra, DOI: 10.1080/03081087.2016.1178209.
Zhu H.H., Patrício P., Chen J.L., Zhang Y.L.: The inverse along a product and its applications. Linear Multilinear Algebra 64, 834–841 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was supported by the National Natural Science Foundation of China (No. 11371089), the Natural Science Foundation of Jiangsu Province (No. BK20141327), and the Foundation of Graduate Innovation Program of Jiangsu Province (No. KYLX_0080).
The third author is supported by the Ministry of Education and Science, Republic of Serbia, grant no. 174007.
Rights and permissions
About this article
Cite this article
Chen, J., Ke, Y. & Mosić, D. The reverse order law of the (b, c)-inverse in semigroups. Acta Math. Hungar. 151, 181–198 (2017). https://doi.org/10.1007/s10474-016-0667-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-016-0667-1