Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T16:31:08.806Z Has data issue: false hasContentIssue false

XV.—On Non-Associative Combinations

Published online by Cambridge University Press:  15 September 2014

I. M. H. Etherington
Affiliation:
Mathematical Institute, University of Edinburgh
Get access

Extract

Numerous combinatory problems arise in connection with a set of elements subject to a non-associative process of composition—let us say of multiplication—commutative or non-commutative.

Non-associative products may be classified according to their shape. By the shape of a product I mean the manner of association of its factors without regard to their identity. Shapes will be called commutative or non-commutative according to the type of multiplication under consideration. Thus if multiplication is non-commutative, the products (AB.C)D and (BA.C)D are distinct but have the same shape, while D(AB.C) has a different shape. The three expressions, however, have the same commutative shape. I confine attention to products (like these) in which the factors are combined only two at a time.

Type
Proceedings
Copyright
Copyright © Royal Society of Edinburgh 1940

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

Binet, M. J., 1839. “Réflexions sur le problème de déterminer le nombre de manières dont une figure rectiligne peut être partagée en triangles au moyens de ses diagonales,” Journ. de Math. (1), vol. iv, pp. 7990.Google Scholar
Catalan, E., 1838. “Note sur une équation aux différences finies,” Journ. de Math., (1) vol. iii, pp. 508516. Also 1839, vol. iv, pp. 91–94, 95–99; 1841, vol. vi, p. 74.Google Scholar
Cayley, A., 1857, 1859. “On the theory of the analytical forms called trees,” Phil. Mag., vol. xiii, pp. 172176; vol. xviii, pp. 374–378. Also 1875, Rep. Brit. Assoc. Adv. Sci., pp. 257–305; 1881, Amer. Journ. Math., vol. iv, pp. 266–268; 1889, Quart. Journ. Pure App. Math., vol. xxiii, pp. 376–378. (Collected Math. Papers, vol. iii, no. 203; iv, 247; ix, 610; xi, 772; xiii, 895.)CrossRefGoogle Scholar
Etherington, I. M. H., 1937. “Non-associate powers and a functional equation,” Math. Gaz., vol. xxi, pp. 3639, 153.CrossRefGoogle Scholar
Netto, E., 1901. Lehrbuch der Combinatorik, Leipzig.Google Scholar
Rodrigues, O., 1838. “Sur le nombre de manières d'effectuer un produit de n facteurs,” Journ. de Math. (1), vol. iii, p. 549.Google Scholar
Schröder, E., 1870. “Vier combinatorische Probleme,” Zeits. Math., vol. xv, pp. 361376.Google Scholar
Wedderburn, J. H. M., 1922. “The functional equation g(x 2) = 2ax + [g(x)]2,” Ann. Math. (2), vol. xxiv, pp. 121140.Google Scholar