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

() |

of

*m*linear equations in

*n*variables, where the and the are positive ordinals, and the variables range over ordinals.

In the particular case *n* = 1 we show that (S) can be reduced to a canonical form having solutions of a relatively simple type, and we use 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].

**[1]**Seymour Sherman,*Some new properties of transfinite ordinals*, Bull. Amer. Math. Soc.**47**(1941), 111–116. MR**0003688**, 10.1090/S0002-9904-1941-07378-7**[2]**W. Sierpiński,*Cardinal and ordinal numbers*, 2nd rev. ed., Monografie Mat., vol. 34, PWN, Warsaw, 1965. MR**33**#2549.

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1976-0419239-0

Article copyright:
© Copyright 1976
American Mathematical Society