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# XV.—On Non-Associative Combinations

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.

References
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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.
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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.)
Etherington, I. M. H., 1937. “Non-associate powers and a functional equation,” Math. Gaz., vol. xxi, pp. 3639, 153.
Netto, E., 1901. Lehrbuch der Combinatorik, Leipzig.
Rodrigues, O., 1838. “Sur le nombre de manières d'effectuer un produit de n facteurs,” Journ. de Math. (1), vol. iii, p. 549.
Schröder, E., 1870. “Vier combinatorische Probleme,” Zeits. Math., vol. xv, pp. 361376.
Wedderburn, J. H. M., 1922. “The functional equation g(x 2) = 2ax + [g(x)]2,” Ann. Math. (2), vol. xxiv, pp. 121140.
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Proceedings of the Royal Society of Edinburgh
• ISSN: 0370-1646
• EISSN: 2059-9153
• URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh
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