Reduction of systems of linear equations in ordinal variables
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- by J. L. Hickman PDF
- Proc. Amer. Math. Soc. 60 (1976), 265-269 Request permission
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].References
- Seymour Sherman, Some new properties of transfinite ordinals, Bull. Amer. Math. Soc. 47 (1941), 111–116. MR 3688, DOI 10.1090/S0002-9904-1941-07378-7 W. Sierpiński, Cardinal and ordinal numbers, 2nd rev. ed., Monografie Mat., vol. 34, PWN, Warsaw, 1965. MR 33 #2549.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 60 (1976), 265-269
- MSC: Primary 04A10
- DOI: https://doi.org/10.1090/S0002-9939-1976-0419239-0
- MathSciNet review: 0419239