Mathematical structures defined by identities

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:373

Article; Proceedings Paper

Petridi, Constantin M.¹ 

[1] Natl & Kapodistrian Univ Athens, Dept Math, Athens 15784, Greece

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

DOI  :  10.1016/j.cam.2019.112547

来源: Elsevier

【 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.

【 授权许可】

Free