arXiv subjects

Petr Vojtěchovský

Publications and source records attributed to Petr Vojtěchovský.

At least 37 records · Page 2Linked to original sources

arXiv2019

Enumeration of involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order

There is a one-to-one correspondence between involutory latin quandles and uniquely $2$-divisible Bruck loops. Bruck loops of odd prime power order are centrally nilpotent. Using linear-algebraic approach to central extensions, we enumerate Bruck loops (and hence involutory latin quandles) of order $3^k$ for $k\le 5$, except for those loops that are central extensions of the cyclic group of order $3$ by the elementary abelian group of order $3^4$. Among the constructed loops there is a Bruck loop of order $3^5$ whose associated $Γ$-loop is not a commutative automorphic loop. We independently enumerate commutative automorphic loops of order $3^k$ for $k\le 5$, with the same omission as in the case of Bruck loops.

math.GR↗

arXiv2019

Involutive latin solutions of the Yang-Baxter equation

Wolfgang Rump showed that there is a one-to-one correspondence between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras in which all left translations $L_x$ are bijections, the squaring map is a bijection, and the identity $(xy)(xz) = (yx)(yz)$ holds. We call these algebras \emph{rumples} in analogy with quandles, another class of binary algebras giving solutions of the Yang-Baxter equation. We focus on latin rumples, that is, on rumples in which all right translations are bijections as well. We prove that an affine latin rumple of order $n$ exists if and only if $n=p_1^{p_1 k_1}\cdots p_m^{p_m k_m}$ for some distinct primes $p_i$ and positive integers $k_i$. A large class of affine solutions is obtained from nonsingular near-circulant matrices $A$, $B$ satisfying $[A,B]=A^2$. We characterize affine latin rumples as those latin rumples for which the displacement group generated by $L_x L_y\inv$ is abelian and normal in the group generated by all translations. We develop the extension theory of rumples sufficiently to obtain examples of latin rumples that are not affine, not even isotopic to a group. Finally, we investigate latin rumples in which the dual identity $(zx)(yx) = (zy)(xy)$ holds as well, and we show, among other results, that the generators $L_x L_y\inv$ of their displacement group have order dividing four.

math.GR↗

arXiv2018

Simply connected latin quandles

A (left) quandle is connected if its left multiplication group acts transitively. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to so-called constant quandle cocycles that form a subset of quandle cocycles. A connected quandle is said to be \emph{simply connected} if it has no nontrivial coverings, or, equivalently, if its second constant cohomology groups are trivial. In this paper we develop a combinatorial approach to constant cohomology. Upon applying our theory, we prove that connected quandles that are affine over cyclic groups are simply connected (extending a result of Graña for quandles of prime size) and that finite doubly transitive quandles of order different from $4$ are simply connected.

math.GR↗

arXiv2017

Automorphisms of dihedral-like automorphic loops

Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops is obtained as follows: Let $m$ be a positive even integer, $G$ an abelian group, and $α$ an automorphism of $G$ that satisfies $α^2=1$ if $m>2$. Then the dihedral-like automorphic loop $\mathrm{Dih}(m,G,α)$ is defined on $\mathbb Z_m\times G$ by $(i,u)(j,v)=(i+j, ((-1)^{j}u+v)α^{ij})$. We prove that two finite dihedral-like automorphic loops $\mathrm{Dih}(m,G,α)$, $\mathrm{Dih}(\overline{m},\overline{G},\overlineα)$ are isomorphic if and only if $m=\overline{m}$, $G=\overline{G}$, and $α$ is conjugate to $\overlineα$ in the automorphism group of $G$. Moreover, for a finite dihedral-like automorphic loop $Q$ we describe the structure of the automorphism group of $Q$ and its subgroup consisting of inner mappings of $Q$.

math.GR↗

arXiv2017

Automorphic loops arising from module endomorphisms

A loop is automorphic if all its inner mappings are automorphisms. We construct a large family of automorphic loops as follows. Let $R$ be a commutative ring, $V$ an $R$-module, $E=\mathrm{End}_R(V)$ the ring of $R$-endomorphisms of $V$, and $W$ a subgroup of $(E,+)$ such that $ab=ba$ for every $a$, $b\in W$ and $1+a$ is invertible for every $a\in W$. Then $Q_{R,V}(W)$ defined on $W\times V$ by $(a,u)(b,v) = (a+b,u(1+b)+v(1-a))$ is an automorphic loop. A special case occurs when $R=k<K=V$ is a field extension and $W$ is a $k$-subspace of $K$ such that $k1\cap W = 0$, naturally embedded into $\mathrm{End}_k(K)$ by $a\mapsto M_a$, $bM_a = ba$. In this case we denote the automorphic loop $Q_{R,V}(W)$ by $Q_{k<K}(W)$. We call the parameters tame if $k$ is a prime field, $W$ generates $K$ as a field over $k$, and $K$ is perfect when $\mathrm{char}(k)=2$. We describe the automorphism groups of tame automorphic loops $Q_{k<K}(W)$, and we solve the isomorphism problem for tame automorphic loops $Q_{k<K}(W)$. A special case solves a problem about automorphic loops of order $p^3$ posed by Jedlička, Kinyon and Vojtěchovský. We conclude the paper with a construction of an infinite $2$-generated abelian-by-cyclic automorphic loop of prime exponent.

math.GR↗

arXiv2017

Code loops in dimension at most 8

Code loops are certain Moufang $2$-loops constructed from doubly even binary codes that play an important role in the construction of local subgroups of sporadic groups. More precisely, code loops are central extensions of the group of order $2$ by an elementary abelian $2$-group $V$ in the variety of loops such that their squaring map, commutator map and associator map are related by combinatorial polarization and the associator map is a trilinear alternating form. Using existing classifications of trilinear alternating forms over the field of $2$ elements, we enumerate code loops of dimension $d=\mathrm{dim}(V)\le 8$ (equivalently, of order $2^{d+1}\le 512$) up to isomorphism. There are $767$ code loops of order $128$, and $80826$ of order $256$, and $937791557$ of order $512$.

math.GR↗

arXiv2017

Bol loops and Bruck loops of order $pq$

Right Bol loops are loops satisfying the identity $((zx)y)x = z((xy)x)$, and right Bruck loops are right Bol loops satisfying the identity $(xy)^{-1} = x^{-1}y^{-1}$. Let $p$ and $q$ be odd primes such that $p>q$. Advancing the research program of Niederreiter and Robinson from $1981$, we classify right Bol loops of order $pq$. When $q$ does not divide $p^2-1$, the only right Bol loop of order $pq$ is the cyclic group of order $pq$. When $q$ divides $p^2-1$, there are precisely $(p-q+4)/2$ right Bol loops of order $pq$ up to isomorphism, including a unique nonassociative right Bruck loop $B_{p,q}$ of order $pq$. Let $Q$ be a nonassociative right Bol loop of order $pq$. We prove that the right nucleus of $Q$ is trivial, the left nucleus of $Q$ is normal and is equal to the unique subloop of order $p$ in $Q$, and the right multiplication group of $Q$ has order $p^2q$ or $p^3q$. When $Q=B_{p,q}$, the right multiplication group of $Q$ is isomorphic to the semidirect product of $\mathbb{Z}_p\times \mathbb{Z}_p$ with $\mathbb{Z}_q$. Finally, we offer computational results as to the number of right Bol loops of order $pq$ up to isotopy.

math.GR↗

arXiv2016

Central and medial quasigroups of small order

We enumerate central and medial quasigroups of order less than $128$ up to isomorphism, with the exception of those quasigroups that are isotopic to $C_4\times C_2^4$, $C_2^6$, $C_3^4$ or $C_5^3$. We give an explicit formula for the number of quasigroups that are affine over a finite cyclic group.

math.GR↗

arXiv2016

Distributive and trimedial quasigroups of order 243

We enumerate three classes of non-medial quasigroups of order $243=3^5$ up to isomorphism. There are $17004$ non-medial trimedial quasigroups of order $243$ (extending the work of Kepka, Bénéteau and Lacaze), $92$ non-medial distributive quasigroups of order $243$ (extending the work of Kepka and Němec), and $6$ non-medial distributive Mendelsohn quasigroups of order $243$ (extending the work of Donovan, Griggs, McCourt, Opršal and Stanovský). The enumeration technique is based on affine representations over commutative Moufang loops, on properties of automorphism groups of commutative Moufang loops, and on computer calculations with the \texttt{LOOPS} package in \texttt{GAP}.

math.GR↗

arXiv2015

Colored graphs without colorful cycles

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation $m\circ n\equiv m+n-2$, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having $2$, $4$, and $5$ vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality.

math.CO↗

arXiv2015

Periods in missing lengths of rainbow cycles

A cycle in an edge-colored graph is said to be rainbow if no two of its edges have the same color. For a complete, infinite, edge-colored graph $G$, define $\mathfrak{S}(G)=\{n\ge 2\;|\;\text{no $n$-cycle of $G$ is rainbow}\}$. Then $\mathfrak{S}(G)$ is a monoid with respect to the operation $n\circ m = n+m-2$, and thus there is a least positive integer $π(G)$, the period of $\mathfrak{S}(G)$, such that $\mathfrak{S}(G)$ contains the arithmetic progression $\{N+kπ(G)\;|\;k\ge 0\}$ for some sufficiently large $N$. Given that $n\in\mathfrak{S}(G)$, what can be said about $π(G)$? Alexeev showed that $π(G)=1$ when $n\ge 3$ is odd, and conjectured that $π(G)$ always divides $4$. We prove Alexeev's conjecture: Let $p(n)=1$ when $n$ is odd, $p(n)=2$ when $n$ is divisible by four, and $p(n)=4$ otherwise. If $2 0$ there are $G$, $H$ such that $4k\in\mathfrak{S}(G)$, $π(G)=2$, and $4k+2\in\mathfrak{S}(H)$, $π(H)=4$.

math.CO↗

arXiv2015

How permutations displace points and stretch intervals

Let $S_n$ be the set of permutations on $\{1,\,\dots,\,n\}$ and $π\in S_n$. Let $\mathrm{d}(π)$ be the arithmetic average of $\{|i-π(i)|;\;1\le i\le n\}$. Then $\mathrm{d}(π)/n\in[0,\,1/2]$, the expected value of $\mathrm{d}(π)/n$ approaches $1/3$ as $n$ approaches infinity, and $\mathrm{d}(π)/n$ is close to $1/3$ for most permutations. We describe all permutations $π$ with maximal $\mathrm{d}(π)$. Let $\mathrm{s}^+(π)$ and $\mathrm{s}^*(π)$ be the arithmetic and geometric averages of $\{|π(i)-π(i+1)|;\;1\le i 1$. We describe all permutations $π$, $σ$ with maximal $\mathrm{s}^+(π)$ and $\mathrm{s}^*(σ)$.

math.CO↗

arXiv2015

Cycle structure of autotopisms of quasigroups and Latin squares

An autotopism of a Latin square is a triple $(α,β,γ)$ of permutations such that the Latin square is mapped to itself by permuting its rows by $α$, columns by $β$, and symbols by $γ$. Let $\mathrm{Atp}(n)$ be the set of all autotopisms of Latin squares of order $n$. Whether a triple $(α,β,γ)$ of permutations belongs to $\mathrm{Atp}(n)$ depends only on the cycle structures of $α$, $β$ and $γ$. We establish a number of necessary conditions for $(α,β,γ)$ to be in $\mathrm{Atp}(n)$, and use them to determine $\mathrm{Atp}(n)$ for $n\le17$. For general $n$ we determine if $(α,α,α)\in\mathrm{Atp}(n)$ (that is, if $α$ is an automorphism of some quasigroup of order $n$), provided that either $α$ has at most three cycles other than fixed points or that the non-fixed points of $α$ are in cycles of the same length.

math.CO↗

arXiv2015

Closest multiplication tables of groups

Suppose that all groups of order $n$ are defined on the same set $G$ of cardinality $n$, and let the \emph{distance} of two groups of order $n$ be the number of pairs $(a,b)\in G\times G$ where the two group operations differ. Given a group $G(\circ)$ of order $n$, we find all groups of order $n$, up to isomorphism, that are closest to $G(\circ)$.

math.GR↗

arXiv2015

Overlapping latin subsquares and full products

We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order $n$ cannot have more than $\frac nm{n \choose h}/{m\choose h}$ subsquares of order $m$, where $h=\lceil(m+1)/2\rceil$. Indeed, the number of subsquares of order $m$ is bounded by a polynomial of degree at most $\sqrt{2m}+2$ in $n$. (b) For all $n\ge5$ there exists a loop of order $n$ in which every element can be obtained as a product of all $n$ elements in some order and with some bracketing.

math.CO↗

arXiv2015

Powers and alternative laws

A groupoid is alternative if it satisfies the alternative laws $x(xy)=(xx)y$ and $x(yy)=(xy)y$. These laws induce four partial maps on $\mathbb{N}^+\times \mathbb{N}^+$, $(r,\,s)\mapsto (2r,\,s-r)$, $(r-s,\,2s)$, $(r/2,\,s+r/2)$, $(r+s/2,\,s/2)$ that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that $n$th powers in a free alternative groupoid on one generator are well-defined if and only if $n\le 5$. We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses.

math.GR↗

arXiv2015

The Moufang loops of order 64 and 81

We classify Moufang loops of order 64 and 81 up to isomorphism, using a linear algebraic approach to central loop extensions. In addition to the 267 groups of order 64, there are 4262 nonassociative Moufang loops of order 64. In addition to the $15$ groups of order $81$, there are $5$ nonassociative Moufang loops of order $81$, $2$ of which are commutative.

math.GR↗

arXiv2015

Explicit constructions of loops with commuting inner mappings

In 2004, Csörgő constructed a loop of nilpotency class three with abelian group of inner mappings. Until now, no other examples were known. We construct many such loops from groups of nilpotency class two by replacing the product $xy$ with $xyh$ in certain positions, where $h$ is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form.

math.GR↗