Some Remarks on Possible Generalized Inverses in Semigroups

Jovan D. Kečkić

Publications de l'Institut Mathématique, _N_S_61 (1997) no. 75, p. 33 .

Résumé

For a given element $a$ of a semigroup $S$ it is possible that the system of equations in $x$: $axa = a$, $ax=xa$ is inconsistent, and that one or both systems $(S_k)$: $a^{k+1}x=a^k$, $ax=xa$ and $(\Sigma_k)$: $ axa=a$, $a^kx=xa^k $ are consistent for some positive integer $k$, in which case they have more than one solution. Some relations between those two systems are established. However, the chief aim of this note is to investigate the possibilities of extending $(S_k)$, by adding new balanced equations, so that this new system has unique solution. It is proved that if the extended system has unique solution, then the generalized inverse of $a$, defined by it, must be the Drazin inverse. It is also shown that the system $(\Sigma_2)$ $\wedge ax^2=x^2a\wedge xax=x$ cannot be extended into a system with unique solution.

Détail

Classification : 20M05