Commutativity in series of ordinals: a study of invariants

Author:
J. L. Hickman

Journal:
Trans. Amer. Math. Soc. **248** (1979), 411-434

MSC:
Primary 04A10

MathSciNet review:
522267

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that two ordinals are additively commutative if and only if they are finite multiples of some given ordinal, and it is very easy to extend this result to any finite sequence of ordinals. However, no necessary and sufficient conditions for the commutativity of a series of ordinals seem to be known when the length of that series is infinite, although sufficient conditions for certain cases have been given by Sierpiński and Ginsburg. In this paper we present such necessary and sufficient conditions. The general problem is split into five distinct cases: those in which the length of the series is a regular initial ordinal, a singular initial ordinal, an infinite, noninitial prime component, an infinite successor ordinal, and an infinite limit ordinal that is not a prime component. These are dealt with respectively in the second through to the sixth sections of the paper, and it turns out that in every case our criteria can be expressed in terms of an ordinal parameter, which is in fact an invariant of the series in question. This concept of invariance is introduced in the first section, which also contains several lemmas and a slight strengthening of the original Sierpiński-Ginsburg result. The final section of this paper differs from the preceding four sections in two aspects. Firstly, the proofs of its two main results are merely sketched, since they contain no arguments that have not previously appeared in some form or other. Secondly, we have not given any explicit determination of the ordinal parameter introduced in this section, since we felt that such a determination would prolong the paper intolerably and encroach upon work done by J. A. H. Anderson: we have therefore simply referred to Anderson's interesting paper.

**[1]**John A. H. Anderson,*The minimum sum of an arbitrary family of ordinals*, J. London Math. Soc. (2)**7**(1974), 429–434. MR**0332485****[2]**S. Ginsburg,*On the distinct sums of 𝜆-type transfinite series obtained by permuting the elements of a fixed 𝜆-type series*, Fund. Math.**39**(1952), 131–132 (1953). MR**0055403****[3]**J. L. Hickman,*A problem on series of ordinals*, Fund. Math.**81**(1973), no. 1, 49–56. Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, I. MR**0332487****[4]**J. L. Hickman,*Concerning the number of sums obtainable from a countable series of ordinals by permutations that preserve the order-type*, J. London Math. Soc. (2)**9**(1974/75), 239–244. MR**0363910****[5]**J. L. Hickman,*Some results on series of ordinals*, Z. Math. Logik Grundlagen Math.**23**(1977), no. 1, 1–18. MR**0485386****[6]**Wacław Sierpiński,*Sur les séries infinies de nombres ordinaux*, Fund. Math.**36**(1949), 248–253 (French). MR**0035807****[7]**-,*Cardinal and ordinal numbers*, Warszawa, 1965.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
04A10

Retrieve articles in all journals with MSC: 04A10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0522267-5

Article copyright:
© Copyright 1979
American Mathematical Society