Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-01T03:50:49.351Z Has data issue: false hasContentIssue false
  • English
  • Français

VI.—Endomorphisms of Abstract Algebras

Published online by Cambridge University Press:  14 February 2012

Trevor Evans
Trevor Evans
Affiliation:
Department of Mathematics, Emory University, Atlanta, Georgia, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

A study is made of the generalization of the entropic law (xy)(zw)—(xz)(yw) to an identity connecting two operations. It is shown that such an identity is equivalent “in the large” to the condition that the set of endomorphisms with respect to one operation is closed with respect to the other. Furthermore, for such entropic operations, each may be regarded as a generalized endomorphism of the other and various generalizations of elementary properties of endomorphisms are obtained. Examples of entropic pairs of operations are quite common in mathematics and a number of these are discussed. An important aspect of the paper is the matrix notationintroduced to facilitate what would otherwise be extremely cumbersome computations with entropic operations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 66 , Issue 1 , 1962 , pp. 53 - 64
Copyright
Copyright © Royal Society of Edinburgh 1962

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References to Literature

[1]Etherington, I. M. H., 1949. “Non-associative arithmetics”, Proc. Roy. Soc. Edin., A, 62, 442–453.Google Scholar
[2]Etherington, I. M. H., 1958. “Groupoids with additive endomorphisms”, Amer. Math. Mon., 65,596–601.CrossRefGoogle Scholar
[3]Evans, Trevor, 1961. “A condition for a group to be commutative”, Amer. Math. Mon., 68, 898–899.CrossRefGoogle Scholar
[4]Evans, Trevor, 1962. “Properties of algebras almost equivalent to identities”, J. Lond. Math. Soc, 37, 53–59.CrossRefGoogle Scholar
[5]Evans, Trevor, “Abstract mean values” (submitted for publication).Google Scholar
[6]Frink, Orrin, 1955. “Symmetric and self-distributive systems”, Amer. Math. Mon., 62, 697–707.CrossRefGoogle Scholar
[7]Murdoch, D. C, 1941. “Structure and abelian quasigroups”, Trans. Amer. Math. Soc, 49, 392–409.CrossRefGoogle Scholar
[8]Stein, S. K., 1957. “On the foundations of quasigroups”, Trans. Amer. Math. Soc, 85, 228–256.CrossRefGoogle Scholar
[9]Toyoda, K., 1941. “On axioms of linear functions”, Proc. Imp. Acad. Tokyo, 17, 221–227.Google Scholar