Solving Linear Systems in Dioids

How can we expect to solve a linear equality system in algebraic structures consisting of a set with two internal laws $\oplus$ and $\otimes$ which are not a priori invertible, i.e. where one cannot always solve a $\oplus$ x = b and a $\otimes$ x = b?

The key idea in the present chapter is to observe that the solution of a “fixed point” type equation such as x = a $\otimes$ x$\oplus$b only requires the existence of the quasi-inverse a* of the element a, defined in the previous chapter as the “limit” of the series: e $\oplus$ a $\oplus$ a² $\oplus$···

It is indeed remarkable that neither the additive inverse nor the multiplicative inverse are needed to compute “(e – a)^–1”!

However, in order to guarantee some form of uniqueness, it will be necessary to work in canonically ordered semirings, i.e. in dioids.

The purpose of this chapter is thus to discuss how to solve linear systems of the fixed point type, which will lead to generalizations of the main known algorithms for solving linear systems in classical linear algebra.

This chapter does not address the problem of solving linear systems of the form A x = b in dioids. This actually relates to residuation theory introduced in Chap. 3, Sect. 8, and involves a concept of generalized pseudo-inverse.