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On direct decompositions. I

Published online by Cambridge University Press:  24 October 2008

A. W. Goldie
A. W. Goldie
Affiliation:
King's CollegeNewcastle
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Extract

It is known that the refinement theorems for direct decompositions, now classical for group theory, are true, under suitable chain conditions, for two large classes of algebras. These are (1) the algebras whose congruences commute and (2) the algebras with a binary operation of a particular type generated by what we shall call a decomposition operator. On the other hand, there exist finite algebras for which the refinement theorems are not true. The problem of characterizing algebras for which the theorems are true arises at once. It is difficult, because the structures of algebras of classes (1) and (2) can be, to a large extent, incompatible. To illustrate this, we mention that any algebra of (2) has a naturally defined centre which is an Abelian group under the binary operation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 1 , January 1952 , pp. 1 - 22
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

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