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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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