Petr Vojtvechovsk'y

activity

20042022

most citedInvolutive latin solutions of the Yang-Baxter equation

15 citations · 61 across the 31 of their papers we have counts for

collaborators

37 papers

math.GR2022

Supernilpotent groups and $3$ -supernilpotent loops

David Stanovský, Petr Vojtěchovský

We find a short equational basis for the variety of $3$ -supernilpotent loops. We also present a conceptually simple proof that $k$ -nilpotence and $k$ -supernilpotence are equivalent…

math.QA2020

Idempotent solutions of the Yang-Baxter equation and twisted group division

David Stanovský, Petr Vojtěchovský

Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the i…

math.GR2020

Subdirect products and propagating equations with an application to Moufang theorem

Aleš Drápal, Petr Vojtěchovský

We introduce the concept of propagating equations and focus on the case of associativity propagating in varieties of loops. An equation $ε$ propagates in an algebra $X$ i…

math.GR2019

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

Izabella Stuhl, Petr Vojtěchovský

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 l…

math.GR2019

Bol loops and Bruck loops of order $pq$ up to isotopism

Petr Vojtěchovský

Let $p > q$ be odd primes. We classify Bol loops and Bruck loops of order $pq$ up to isotopism. When $q$ does not divide $p^{2} - 1$ , the only Bol loop (and hence the only Bruck loop) of…

math.GT2019

Set-theoretic Yang-Baxter (co)homology theory of involutive non-degenerate solutions

Józef H. Przytycki, Petr Vojtěchovský, Seung Yeop Yang

W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang-Baxter equation and Rump right quasigroups. J. S. Carter,…

math.GR2019★ 15 cited

Involutive latin solutions of the Yang-Baxter equation

Marco Bonatto, Michael Kinyon, David Stanovský +1

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…

math.GR2017

Code loops in dimension at most 8

E. A. O'Brien, Petr Vojtěchovský

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 preci…

math.GR2017

Automorphic loops arising from module endomorphisms

Alexandr Grishkov, Marina Rasskazova, Petr Vojtěchovský

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…

math.GR2016★ 6 cited

Bol loops and Bruck loops of order $pq$

Michael Kinyon, Gábor P. Nagy, Petr Vojtěchovský

Right Bol loops are loops satisfying the identity $((z x) y) x = z ((x y) x)$ , and right Bruck loops are right Bol loops satisfying the identity $(x y)^{- 1} = x^{- 1} y^{- 1}$ . Let $p$ and $…

math.GR2016

Simply connected latin quandles

Marco Bonatto, Petr Vojtěchovský

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 constan…

math.GR2015★ 4 cited

Central and medial quasigroups of small order

David Stanovský, Petr Vojtěchovský

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_…

math.GR2015

Commutator theory for loops

David Stanovský, Petr Vojtěchovský

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 o…

math.GR2015★ 1 cited

Small loops of nilpotency class three with commutative inner mapping groups

Aleš Drápal, Petr Vojtěchovský

Groups with commuting inner mappings are of nilpotency class at most two, but there exist loops with commuting inner mappings and of nilpotency class higher than two, called loops…

math.NT2015

Symmetric multilinear forms and polarization of polynomials

Aleš Drápal, Petr Vojtěchovský

We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ varia…

math.GR2015★ 2 cited

Powers and alternative laws

Nicholas Ormes, Petr Vojtěchovský

A groupoid is alternative if it satisfies the alternative laws $x (x y) = (xx) y$ and $x (y y) = (x y) y$ . These laws induce four partial maps on $N^{+} \times N^{+}$ , $(r,\,s)\m…

math.CO2015

Periods in missing lengths of rainbow cycles

Petr Vojtěchovský

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…

math.GR2015★ 5 cited

Three lectures on automorphic loops

Petr Vojtěchovský

These notes accompany a series of three lectures on automorphic loops to be delivered by the author at Workshops Loops '15 (Ohrid, Macedonia, 2015). Automorphic loops are loops in…

math.GR2007

The varieties of quasigroups of Bol-Moufang type: An equational reasoning approach

J. D. Phillips, Petr Vojtěchovský

A quasigroup identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, the order in which the variables a…

math.GR2007

Toward the classification of Moufang loops of order 64

Petr Vojtěchovský

We show how to obtain all nonassociative Moufang loops of order less than 64 and 4262 nonassociative Moufang loops of order 64 in a unified way. We conjecture that there are no oth…

math.GR2007

C-loops: An introduction

J. D. Phillips, Petr Vojtěchovský

C-loops are loops satisfying $x (y (y z)) = ((x y) y) z$ . They often behave analogously to Moufang loops and they are closely related to Steiner triple systems and combinatorics. We initia…

math.GR2007

Moufang loops that share associator and three quarters of their multiplication tables

Aleš Drápal, Petr Vojtěchovský

Two constructions due to Drápal produce a group by modifying exactly one quarter of the Cayley table of another group. We present these constructions in a compact way, and generali…

math.GR2007

A class of Bol loops with a subgroup of index two

Petr Vojtěchovský

Let $G$ be a finite group and $C_{2}$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x, y) \mapsto (x^{i} y^{j})^{k}$ , where $i$ , $j$ , $k \in {- 1, 1}$ . Define a new m…

math.GR2007

Octonions, simple Moufang loops and triality

Gábor P. Nagy, Petr Vojtěchovský

Nonassociative finite simple Moufang loops are exactly the loops constructed by Paige from Zorn vector matrix algebras. We prove this result anew, using geometric loop theory. In o…

Supernilpotent groups and 3-supernilpotent loops

Idempotent solutions of the Yang-Baxter equation and twisted group division

Subdirect products and propagating equations with an application to Moufang theorem

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

Bol loops and Bruck loops of order pq up to isotopism

Set-theoretic Yang-Baxter (co)homology theory of involutive non-degenerate solutions

Involutive latin solutions of the Yang-Baxter equation

Code loops in dimension at most 8

Automorphic loops arising from module endomorphisms

Bol loops and Bruck loops of order pq

Simply connected latin quandles

Central and medial quasigroups of small order

Commutator theory for loops

Small loops of nilpotency class three with commutative inner mapping groups

Symmetric multilinear forms and polarization of polynomials

Powers and alternative laws

Periods in missing lengths of rainbow cycles

Three lectures on automorphic loops

The varieties of quasigroups of Bol-Moufang type: An equational reasoning approach

Toward the classification of Moufang loops of order 64

C-loops: An introduction

Moufang loops that share associator and three quarters of their multiplication tables

A class of Bol loops with a subgroup of index two

Octonions, simple Moufang loops and triality

Supernilpotent groups and $3$ -supernilpotent loops

Bol loops and Bruck loops of order $pq$ up to isotopism

Bol loops and Bruck loops of order $pq$