(b, c)-inverse, inverse along an element, and the Schützenberger category of a semigroup

Document Type : Research Paper


Laboratoire Modal’X, Université Paris Nanterre, France


We prove that the (b, c)-inverse and the inverse along an element
in a semigroup are actually genuine inverse when considered as morphisms
in the Schützenberger category of a semigroup. Applications to the Reverse
Order Law are given.


[1] Chen, J., Ke, Y., and Mosić, D., The reverse order law of the (b, c)−inverse in semigroups, Acta Math. Hungar. 151(1) (2017), 181-198.
[2] Costa, A. and Steinberg, B., The Schützenberger category of a semigroup, Semigroup Forum 91(3) (2015), 543-559.
[3] Diekert, V., Kufleitner, M., and Steinberg, B., The Krohn-Rhodes theorem and local divisors, Fundamenta Informaticae 116(1-4) (2012), 65-77.
[4] Drazin, M.P., A class of outer generalized inverses, Linear Algebra Appl. 436(7) (2012), 1909-1923.
[5] Green, J.A., On the structure of semigroups, Ann. of Math. (1951), 163-172.
[6] Greville, T.N.E., Note on the generalized inverse of a matrix product, Siam Review 8(4) (1966), 518-521.
[7] Hollings, C., The Ehresmann-Schein-Nambooripad Theorem and its successors, Eur. J. Pure Appl. Math. 5(4) (2012), 414-450.
[8] Mary, X., On generalized inverses and Green’s relations, Linear Algebra Appl. 434(8) (2011), 1836-1844.
[9] Mary, X. and Patrício, P., Generalized inverses modulo 𝓗 in semigroups and rings, Linear Multilinear Algebra 61(8) (2013), 1130-1135.
[10] Miller, D.D. and Clifford, A.H., Regular 𝓓-classes in semigroups, Tran. Amer. Math. Soc. 82(1) (1956), 270-280.
[11] Mitsch, H. A natural partial order for semigroups, Proc. Amer. Math. Soc. 97(3) (1986), 384-388.
[12] Nambooripad, K.S.S., ‘’Theory of Cross-connections’’, Centre for Mathematical Sciences, 1994.
[13] Schützenberger, MP. D̅-représentation des demi-groupes, C. R. Acad. Sci. Paris. 244(15) (1957), 1994-1996.