Commutator theory for racks and quandles
نویسندگان
چکیده
We adapt the commutator theory of universal algebra to particular setting racks and quandles, exploiting a Galois connection between congruences certain normal subgroups displacement group. Congruence properties, such as abelianness centrality, are reflected by corresponding relative groups, global solvability nilpotence, properties whole To show new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory order $2^k$, no latin $\equiv2\pmod4$), non-colorability theorem (knots with trivial Alexander polynomial not colorable solvable quandles; particular, finite quandles), strengthening Glauberman's results on Bruck loops odd order.
منابع مشابه
Extensions of Racks and Quandles
A rack is a set equipped with a bijective, self-right-distributive binary operation, and a quandle is a rack which satisfies an idempotency condition. In this paper, we introduce a new definition of modules over a rack or quandle, and show that this definition includes the one studied by Etingof and Graña [9] and the more general one given by Andruskiewitsch and Graña [1]. We further show that ...
متن کاملOn Finite Racks and Quandles
We revisit finite racks and quandles using a perspective based on permutations which can aid in the understanding of the structure. As a consequence we recover old results and prove new ones. We also present and analyze several examples.
متن کاملEnumerating Finite Racks, Quandles and Kei
A rack of order n is a binary operation B on a set X of cardinality n, such that right multiplication is an automorphism. More precisely, (X,B) is a rack provided that the map x 7→ x B y is a bijection for all y ∈ X, and (x B y) B z = (x B z) B (y B z) for all x, y, z ∈ X. The paper provides upper and lower bounds of the form 2cn 2 on the number of isomorphism classes of racks of order n. Simil...
متن کاملYang-Baxter deformations of quandles and racks
Given a rack Q and a ring A , one can construct a Yang-Baxter operator cQ : V ⊗ V → V ⊗ V on the free A-module V = AQ by setting cQ(x ⊗ y) = y ⊗ x y for all x, y ∈ Q . In answer to a question initiated by D.N.Yetter and P.J. Freyd, this article classifies formal deformations of cQ in the space of Yang-Baxter operators. For the trivial rack, where x = x for all x, y , one has, of course, the cla...
متن کاملCommutator Theory for Loops
Using the Freese-McKenzie commutator theory for congruence modular varieties as the starting point, we develop commutator theory for the variety of loops. The fundamental theorem of congruence commutators for loops relates generators of the congruence commutator to generators of the total inner mapping group. We specialize the fundamental theorem into several varieties of loops, and also discus...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of The Mathematical Society of Japan
سال: 2021
ISSN: ['1881-1167', '0025-5645']
DOI: https://doi.org/10.2969/jmsj/83168316