Reduction of systems of linear equations in ordinal variables

Abstract:

In this note we are concerned with a general finite system \begin{equation}\tag {$S$} \sum \limits _{i = 0}^{n - 1} {{x_i}{\alpha _{ji}} = {\beta _j};\quad j < m,}\end{equation} of m linear equations in n variables, where the ${\alpha _{ji}}$ and the ${\beta _j}$ are positive ordinals, and the variables ${x_i}$ range over ordinals. In the particular case n = 1 we show that (S) can be reduced to a canonical form $({{\text {S}}^\ast })$ having solutions of a relatively simple type, and we use $({{\text {S}}^\ast })$ to obtain the solution-set of (S). In the general case we show that (S) can be reduced to a finite sequence of single-variable systems, and again obtain the solution-set of (S) in terms of the solution-sets of these simpler systems. We assume a knowledge of the elementary theory of ordinal arithmetic, such as may be found for example in [2].