Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T02:53:54.935Z Has data issue: false hasContentIssue false
  • English
  • Français

XI.—Theory of Indices for Non-Associative Algebra*

Published online by Cambridge University Press:  14 February 2012

I. M. H. Etherington
I. M. H. Etherington
Affiliation:
Mathematical Institute, University of Edinburgh.
Get access Rights & Permissions [Opens in a new window]

Synopsis

The logarithmetic L of a non-associative algebra or class of algebras S has been previously defined as the arithmetic of the indices of powers of the general element when indices are added (non-associatively) and multiplied by certain conventions similar to those of ordinary algebra. With respect to addition, L is a homomorphic image of the “most general” logarithmetic B, the free additive groupoid with one generator 1, and in the case of algebras of one operation is essentially the same as the free algebra in one variable on S. The definition is now extended so that L is defined when S is any subset of an algebra or class of subsets of algebras, with the result that every homomorph of B is a logarithmetic ; but a distinction has then to be drawn between closed logarithmetics in which as before both addition and multiplication are defined, and other logarithmetics in which there is only addition. L is its own logarithmetic (taken with respect to addition) only if L is closed. For subsets of palintropic algebras, L is necessarily closed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 64 , Issue 2 , 1954 , pp. 150 - 160
Copyright
Copyright © Royal Society of Edinburgh 1954

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

Birkhoff, G., 1935. “On the structure of abstract algebras”, Proc. Camb. Phil. Soc., 31, 433–454.CrossRefGoogle Scholar
Birkhoff, G., 1945. “Universal algebra”, Proc. First Canadian Mathematical Congress (Montreal), 310–326.Google Scholar
Birkhoff, G., 1948. Lattice Theory. New York.Google Scholar
Cayley, A., 1859. “On the theory of the analytical forms called trees”, Phil. Mag., 18, 374–378; Collected Math. Papers, 4, No. 247.CrossRefGoogle Scholar
Etherington, I. M. H., 1939. “On non-associative combinations”, Proc. Roy. Soc. Edin., 59, 153–162.CrossRefGoogle Scholar
Etherington, I. M. H., 1940, 1945. “Commutative train algebras of ranks 2 and 3”, J. Lond. Math. Soc., 15, 136–149; 20, 238.CrossRefGoogle Scholar
Etherington, I. M. H., 1941. “Some non-associative algebras in which the multiplication of indices is commutative”, J. Lond. Math. Soc., 16, 48–55.CrossRefGoogle Scholar
Etherington, I. M. H., 1945. “Transposed algebras”, Proc Edin. Math. Soc.. (2), 7, 104–121.CrossRefGoogle Scholar
Etherington, I. M. H., 1949. “Non-associative arithmetics”, Proc. Roy. Soc. Edin., A, 62, 442–453.Google Scholar
Etherington, I. M. H., 1951. “Non-commutative train algebras of ranks 2 and 3”, Proc. Lond. Math. Soc. (2), 52, 241–252.Google Scholar
Etherington, I. M. H., and Erdélyi, A., 1941. “Some problems of non-associative combinations”, Edin. Math, Notes, No. 32, 1–6, 7–12.Google Scholar
Murdoch, D. C., 1939. “Quasi-groups which satisfy certain generalized associative laws”, Amer. J. Math., 61, 509–522.CrossRefGoogle Scholar
Murdoch, D. C., 1941. “Structure of abelian quasi-groups”, Trans. Amer. Math. Soc., 49, 392–409.CrossRefGoogle Scholar
Popova, H., 1952, 1953. “Logarithmétiques des quasi-groupes finis” and other titles, C.R. Acad. Sci. Pari., 234, 1936–1937, 2582–2583; 235, 1360–1362, 1589–1591; 236, 769–771, 1220–1222.Google Scholar
Popova, H., 1954. “Logarithmetics of finite quasigroups. I”, Proc. Edin. Math. Soc. (2), 9, 74–81.CrossRefGoogle Scholar
Robinson, A., 1949. “On non-associative systems”, Proc. Edin. Math. Soc. (2), 8, 111–118.CrossRefGoogle Scholar