Abstract
The problem of determining the conditions that must be imposed upon a system having a single associative and commutative operation in order to obtain unique factorization into irreducibles has been studied by A. H. Clifford [1],1 König [1], and Ward [2]. The more general problem of determining similar conditions for the non-commutative case has been treated by M. Ward [1], However, the conditions given by Ward are more stringent than those satisfied by actual instances of non-commutative arithmetic, for example, quotient lattices and non-commutative polynomial theory (Ore [1, 2]). Moreover, in both of these instances the factorization is unique only up to a similarity relation, and instead of a single operation of multiplication the additional operations G. C. D. and L. C. M. are involved.2 Accordingly, we shall concern ourselves with the arithmetic of a non-commutative multiplication defined over a lattice.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Garrett Birkhoff. On combination of subalgebras, Proc. Cambridge Phil. Soc., vol. 29(1933), pp. 441–464.
A.H. Clifford. Arithmetic and ideal theory of abstract multiplication, Bull. Am. Math. Soc., vol. 40(1934), pp. 326–330.
J. König. Algebraischen Grossen, Leipzig, 1903, Chapter I.
O. Ore. Abstract algebra, I, II, Annals of Math., vol. 36(1935), pp. 406–437; vol. 37(1936), pp. 265-292.
O. Ore. Theory of non-commutative polynomials, Annals of Math., vol. 34(1933), pp. 480–508.
M. Ward. Postulates for an abstract arithmetic, Proc. Natl. Acad. Sci., vol. 14(1928), pp. 907–911.
M. Ward. Conditions for factorization in a set closed under a single operation, Annals of Math., vol. 36(1933), pp. 36–39.
M. Ward. Structure residuation, Annals of Math., vol. 39 (1938), pp. 558–568.
M. Ward and R.P. Dilworth. Residuated lattices, Proc. Natl. Acad. Sci., vol. 24(1938), pp. 162–165.
M. Ward and R.P. Dilworth. Residuated lattices, to appear in the Trans, of the Amer. Math. Soc.
J.H.M. Wedderburn. Non-commutative domains of integrity, Journal für Math., vol. 167(1931), pp. 129–141.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer Science+Business Media New York
About this chapter
Cite this chapter
Dilworth, R.P. (1990). Non-Commutative Arithmetic. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_34
Download citation
DOI: https://doi.org/10.1007/978-1-4899-3558-8_34
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4899-3560-1
Online ISBN: 978-1-4899-3558-8
eBook Packages: Springer Book Archive