Algebraic analysis in structures with the Kaplansky–Jacobson property
In 1950 N. Jacobson proved that if $u$ is an element of a ring with unit such that $u$ has more than one right inverse, then it has infinitely many right inverses. He also mentioned that I. Kaplansky proved this in another way. Recently, K. P. Shum and Y. Q. Gao gave a new (non-constructive) proof of the Kaplansky–Jacobson theorem for monoids admitting a ring structure. We generalize that theorem to monoids without any ring structure and we show the consequences of the generalized Kaplansky–Jacobson theorem for the theory of linear operators, and even for the classical Calculus. In order to do that, we recall some multiplicative systems, called pseudocategories, very useful in the algebraic theory of perturbations of linear operators. In the second part of the paper, basing on the Kaplansky–Jacobson theorem, we show how to use the above mentioned structures for building Algebraic Analysis of linear operators over a class of linear spaces. We also define (non-linear) logarithmic and antilogarithmic mappings on these structures.