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On Non-Associative Systems

Published online by Cambridge University Press:  20 January 2009

A. Robinson
A. Robinson
Affiliation:
College of Aeronautics, Cranfield, Bletchley, Bucks.
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The present paper is concerned with the “logarithmetic”, or arithmetic of shapes of non-associative combinations as defined by Etherington in ref. (1). The shape of a non-associative product is defined as “the manner of association of its factors without regard to their identity”. Thus, for a binary non-communicative operation, the products ((AB)C)D and ((BA)C)D and ((AA)A)A all have the same shape, while D((AB)C) has a different shape. The sum of the two shapes a and b is defined as the shape of the product of two expressions, of shapes a and b respectively, in the original system of non-associative combination. The product of two shapes a and b is defined as the shape of any expression obtained by replacing every factor in an expression of shape b by an expression of shape a. It is readily shown that these definitions are unambiguous.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 8 , Issue 3 , December 1949 , pp. 111 - 118
Copyright
Copyright © Edinburgh Mathematical Society 1949

References

REFERENCES

(1)Etherington, I. M. H., “On non-associative combinations,” Proc. Roy. Soc. Edinburgh, LIX (1939), 153–162.Google Scholar
(2)Cayley, A., “On the theory of the analytical forms called trees,” Phil. Mag., XIII (1857), 172–176.CrossRefGoogle Scholar