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XLVII.—Non-Associative Arithmetics*

Published online by Cambridge University Press:  14 February 2012

I. M. H. Etherington
I. M. H. Etherington
Affiliation:
University of Edinburgh
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Introduction and summary

The systems of “partitive numbers” introduced in this paper differ from ordinary number systems in being subject to non-associative addition. They are intended primarily to serve as the indices of powers in algebraic systems having non-associative multiplication, or as the coefficients of multiples in systems with non-associative addition, but are defined more generally than is probably necessary for these purposes. They are essentially the same as root-trees (Setzbäume) with non-branching knots other than terminal knots ignored, with operations of addition and multiplication defined.

Partitive numbers are of two kinds, partitioned cardinals and partitioned serials, defined respectively as the partition-types of repeatedly partitioned classes and series. For each kind, multiplication is binary (i.e. any ordered pair has a unique product) and associative. Addition is in general a free operation (i.e. the summands are not limited to two, and indeed, assuming the multiplicative axiom, may form an infinite class or series); but it is non-associative, which means that for example a + b + c (involving one operation of addition) is distinguished from (a + b) + c and a + (b + c) (involving two operations). A one-sided distributive law is obeyed:

Partitioned cardinals are commutative in addition.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 62 , Issue 4 , 1949 , pp. 442 - 453
Copyright
Copyright © Royal Society of Edinburgh 1949

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References

References to Literature

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