## Some remarks on possible generalized inverses in semigroups.(English)Zbl 0901.20045

For a given element $$a$$ of a semigroup and for some positive integer $$k$$ the systems of equations in $$x$$ $$(S_k)$$: $$a^{k+1}x=a^k$$, $$ax=xa$$ and $$(\Sigma_k)$$: $$axa=a$$, $$a^kx=xa^k$$ are considered and some relations between these two systems are established. If $$k=1$$, both systems reduce to the well known system $$axa=a$$, $$ax=xa$$. The main result is: If the system $$(S_k)$$ is consistent, then it can be extended by adding new balanced equations so that the new system has a unique solution. The solution is the Drazin inverse of $$a$$. It is also shown that the system $$(\Sigma_2)$$, $$ax^2=x^2a$$, $$xax=x$$ cannot be extended to a system with unique solution.

### MSC:

 20M05 Free semigroups, generators and relations, word problems 15A09 Theory of matrix inversion and generalized inverses
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