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On complete atomic proper relation algebras

Published online by Cambridge University Press:  12 March 2014

Frank Harary
Frank Harary*
Affiliation:
University of Michigan
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Extract

The object of this note is to classify the isomorphism types of all complete atomic proper relation algebras. In particular we find that the number of abstractly distinct complete atomic proper finite relation algebras of 2m elements is equal to the number of distinct partitions of the positive integer m into summands which are perfect squares.

A relation algebra is called complete atomic if its Boolean algebra is complete atomic. A proper relation algebra is a relation algebra whose elements are relations. Lyndon has recently proved, by constructing an appropriate finite relation algebra, that not every complete atomic relation algebra is isomorphic to a proper relation algebra.

Henceforth, let us consider a complete atomic proper relation algebra. The atoms {〈a, b〉}, {〈c, d〉} will be called connected if and only if {a, b} ⋂ {c, d} is not empty. Let the relation of connectedness over the set B of all atoms be denoted by ~. For brevity let B = {b1,b2, …}. The relation ~ is obviously reflexive and symmetric. But ~ is not transitive, since a, b, c, d distinct implies {〈a, 0〉} ~ {〈b, c〉}, {〈b, c〉} ~ {〈c, d〉}, and {〈a, b〉} ≁ {〈c, d〉}.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 15 , Issue 3 , September 1950 , pp. 197 - 198
Copyright
Copyright © Association for Symbolic Logic 1950

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References

1 For definition of relation algebra, see Jonsson, B. and Tarski, A.—Representation problems for relation algebras, Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 80 and 1192, Abstract 89Google Scholar.

2 Terminology from ibid.

3 Lyndon, R. C., The representation of algebras of relations, Bulletin of the American Mathematical Society, vol. 55 (1949), pp. 513–514, Abstract 234Google Scholar.

4 This remark is due to the referee.

5 (Added August 3, 1950.) I have just noticed that the above theorems and remarbs are valid only for those complete atomic proper relation algebras in which every atom is a singleton—i. e., a relation consisting of exactly one ordered couple. The isomorphism types of all complete atomic proper relation algebras are now being investigated and will be presented later.