Mathematical structures defined by identities

[摘要] In this work we propose that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally irreducible identity between two n-iterates of the operation, it is established that the frequency of such algebras goes to zero. This is proved by a suitable ordering and labeling of the expressions (words) of the corresponding free algebra and the formation of a series of tableaux whose entries are the labels. The tableaux reveal surprising symmetry properties, stated in terms of the Catalan numbers 1/n+1 2nn and their partitions. Class numbers depending on the tableaux are calculated tor all algebras of order n = 3 and partially for n = 4. Certain class numbers are invariants in the sense that for algebras of same order they are equal. (C) 2019 Elsevier B.V. All rights reserved.

[发布日期] 2020-08-01

[效力级别]  Proceedings Paper

[关键词] Reducible or irreducible identities;Catalan numbers;Incidence matrices;Inclusion-exclusion principle;Groups