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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Reduction of systems of linear equations in ordinal variables

Author: J. L. Hickman
Journal: Proc. Amer. Math. Soc. 60 (1976), 265-269
MSC: Primary 04A10
MathSciNet review: 0419239
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Abstract: In this note we are concerned with a general finite system

$\displaystyle \sum\limits_{i = 0}^{n - 1} {{x_i}{\alpha _{ji}} = {\beta _j};\quad j < m,}$ ($ S$)

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 [Enhancements On Off] (What's this?)

  • [1] S. Sherman, Some new properties of transfinite ordinals, Bull. Amer. Math. Soc. 47 (1941), 111-116. MR 2, 255. MR 0003688 (2:255j)
  • [2] W. Sierpiński, Cardinal and ordinal numbers, 2nd rev. ed., Monografie Mat., vol. 34, PWN, Warsaw, 1965. MR 33 #2549.

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Article copyright: © Copyright 1976 American Mathematical Society

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