Associativity conditions for linear quasigroups and equivalence relations on binary trees

We characterise the bracketing identities satisfied by linear quasigroups with the help of certain equivalence relations on binary trees that are based on the left and right depths of the leaves modulo some integers. The numbers of equivalence classes of $n$-leaf binary trees are variants of the Catalan numbers, and they form the associative spectrum (a kind of measure of non-associativity) of a quasigroup.Comment: 32 page

Colorful Associahedra and Cyclohedra

Every n-edge colored n-regular graph G naturally gives rise to a simple abstract n-polytope, the colorful polytope of G, whose 1-skeleton is isomorphic to G. The paper describes colorful polytope versions of the associahedron and cyclohedron. Like their classical counterparts, the colorful associahedron and cyclohedron encode triangulations and flips, but now with the added feature that the diagonals of the triangulations are colored and adjacency of triangulations requires color preserving flips. The colorful associahedron and cyclohedron are derived as colorful polytopes from the edge colored graph whose vertices represent these triangulations and whose colors on edges represent the colors of flipped diagonals.Comment: 21 pp, to appear in Journal Combinatorial Theory

On free quasigroups and quasigroup representations

This work consists of three parts. The discussion begins with \emph{linear quasigroups}. For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $\rho$ and $\lambda$ giving a (nonassociative) multiplication $x\cdot y=x^\rho+y^\lambda$. If $S$ is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional $S$-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for $\mathbb{Z}$-linear quasigroups. We consider the extent to which ordinary characters classify $\mathbb{Z}$-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic $\mathbb{Z}$-linear quasigroups with the same ordinary character. For a subclass of $\mathbb{Z}$-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on $\mathbb{Z}$-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy. The story progresses with a representation of the free quasigroup on a single generator. This provides the motivation behind the study of \emph{peri-Catalan numbers}. While Catalan numbers index the number of length $n$ magma words in a single generator, peri-Catalan numbers index the number of length $n$ reduced form quasigroup words in a single generator. We derive a recursive formula for the $n$-th peri-Catalan number. This is a new sequence in that it is not on the Online Encyclopedia of Integer Sequences

Donaghey's transformation: carousel effects and tame components

Изучается динамическая система, действующая на комбинаторных интерпретациях чисел Каталана. Описаны два новых карусельных эффекта (для триад и ростков), первый из которых использован для перехода от локального свойства деревьев (отсутствие триад определённых типов) к некоторым глобальным свойствам орбит. Показано, что отмеченный эффект носит массовый характер. Построено несколько новых ручных классов орбит, в том числе не только с постоянными длинами орбит, но и с растущими как 0(n2). Решены соответствующие перечислительные задачи

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework which is presented step-by-step with examples throughout. In this second part of two papers, we give the general categorical formulation