arXiv subjects

Premysl Jedlicka

Publications and source records attributed to Premysl Jedlicka.

7 recordsLinked to original sources

arXiv2024

Non-involutive solutions of the Yang-Baxter equation of multipermutation level 2

We study non-degenerate set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2 which are not 2-reductive. We describe an effective way of constructing such solutions using square-free 2-reductive solutions and two bijections. We present an algorithm how to obtain all such finite solutions, up to isomorphism. Using this algorithm, we enumerate all solutions of multipermutation level 2 up to size 6.

math.RA↗

arXiv2024

Diagonals of solutions of the Yang-Baxter equation

We study the diagonal mappings in non-involutive set-theoretic solutions of the Yang-Baxter equation. We show that, for non-degenerate solutions, they are commuting bijections. This gives the positive answer to the question: ``Is every non-degenerate solution bijective?'' of Ced\'{o}, Jespers and Verwimp. Additionally, we show that for a subclass of solutions called k-permutational, only one-sided non-degeneracy suffices to prove that one of the diagonal mappings is invertible. We also present an equational characterization of multipermutation solutions and extend results of Rump and Gateva-Ivanova about decomposability to non-involutive case. In particular, we show that each, not necessarily involutive, square-free multipermutation solution of finite level and arbitrary cardinality, is always decomposable.

math.RA↗

arXiv2017

Subdirectly irreducible medial quandles

We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation of all non-connected subdirectly irreducible medial quandles.

math.RA↗

arXiv2016

Free medial quandles

This paper brings the construction of free medial quandles as well as free $n$-symmetric medial quandles and free $m$-reductive medial quandles.

math.RA↗

arXiv2011

Nilpotency in automorphic loops of prime power order

A loop is automorphic if its inner mappings are automorphisms. Using so-called associated operations, we show that every commutative automorphic loop of odd prime power order is centrally nilpotent. Starting with anisotropic planes in the vector space of $2\times 2$ matrices over the field of prime order $p$, we construct a family of automorphic loops of order $p^3$ with trivial center.

math.GR↗

arXiv2010

Constructions of commutative automorphic loops

A loop whose inner mappings are automorphisms is an \emph{automorphic loop} (or \emph{A-loop}). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate the classification of commutative A-loops of small orders and also of order $p^3$, where $p$ is a prime.

math.GR↗

arXiv2010

The Structure of Commutative Automorphic Loops

An \emph{automorphic loop} (or \emph{A-loop}) is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and $(xy)^{-1} = x^{-1}y^{-1}$ holds. Let $Q$ be a finite commutative A-loop and $p$ a prime. The loop $Q$ has order a power of $p$ if and only if every element of $Q$ has order a power of $p$. The loop $Q$ decomposes as a direct product of a loop of odd order and a loop of order a power of 2. If $Q$ is of odd order, it is solvable. If $A$ is a subloop of $Q$ then $|A|$ divides $|Q|$. If $p$ divides $|Q|$ then $Q$ contains an element of order $p$. If there is a finite simple nonassociative commutative A-loop, it is of exponent 2.

math.GR↗