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A note on train algebras

Published online by Cambridge University Press:  20 January 2009

Victor M. Abraham
Victor M. Abraham
Affiliation:
School of Mathematics, Statistics and Computing, Thames Polytechnic, London and Department of Statistics, Birkbeck College, London
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Train algebras were first introduced by Etherington in (1) and proved very useful in dealing with problems in mathematical genetics. The types of algebras which arose were commutative, non-associative and finite-dimensional. It proved convenient in the general theory to regard them as defined over the complex numbers. We remind the reader of some basic definitions. A baric algebra is one which admits a non-trivial homomorphism into its coefficient field K. A (principal) train algebra is baric and has a rank equation in which the coefficients of a general element x depend only on its baric value, generally called the weight of x. A special train algebra (STA) is a baric algebra in which the nilideal is nilpotent and all its right powers are ideals; the nilideal being the set of elements of A of weight zero. In (2) Etherington showed that in a baric algebra one can always take a very simple basis consisting of a distinguished element of unit weight and all other basis elements of weight zero.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 1 , March 1976 , pp. 53 - 58
Copyright
Copyright © Edinburgh Mathematical Society 1976

References

REFERENCES

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