Publications of Premysl Jedlicka

Disclaimer: This page contains articles in the submit form. Any changes made after (e.g. final corrections) are usually not reflected.

		Journal of Algebra 466 (2016) 204–207
	Abstract:	The equational variety of quasigroups is defined by six identities, called Birkhoff's identities. It is known, that only four of them suffice to define the variety; actually, there are nine different combinations of four Birkhoff's identities defining quasigroups, other four combinations define larger varieties and it was open whether the remaining two cases define quasigroups or larger classes. We solve the question here constructing examples of algebras that are not quasigroups and satisfy the open cases of Birkhoff's identities.
	Keywords:	Quasigroup; division groupoid; cancellative groupoid; parastrophe [PDF]

		Quasigroups and Related Systems 23 (2015) 237–242
	Abstract:	In this paper we study automorphic loops of exponent 2 which are semidirect products of the Klein group with an elementary abelian group. It turns out that they fall into two classes: extensions of index 2 and extension using a symmetric bilinear form.
	Keywords:	automorphic loop, semidirect product, middle nucleus, exponent 2 [PDF]

		Comment. Math. Univ. Carolin. 55,4 (2014), 447–456
	Abstract:	We analyze semidirect extensions of middle nuclei of commutative automorphic loops. We find less complicated conditions for the semidirect construction when the middle nucleus is an odd order abelian group. We then use the description to study extensions of orders 3 and 5.
	Keywords:	automorphic loop, semidirect product, middle nucleus, cyclic group [PDF]

	with Jan Hora	Journal of Algebra and its Applications 13,1 (2014), 15 pages
Abstract:	Automorphic loops are loops where all inner mappings are automorphisms. We study when a semidirect product of two abelian groups yields a commutative automorphic loop such that the normal subgroup lies in the middle nucleus. With this description at hand we give some examples of such semidirect products.
Keywords:	commutative automorphic loop, semidirect product [PDF]

	with Michael K. Kinyon and Petr Vojtěchovský	Journal of Algebra 350 (2012), no. 1, 64–76
Abstract:	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 2×2 matrices over the field of prime order p, we construct a family of automorphic loops of order p³ with trivial center.
Keywords:	automorphic loop, commutative automorphic loop, A-loop, central nilpotency [PDF]

		Comment. Math. Univ. Carolin. 51 (2010), no. 2, 253–261
	Abstract:	Using a construction of commutative loops with metacyclic inner mapping group and trivial center described by A. Drápal, we enumerate presumably all such loops of order pq, for p and q primes.
	Keywords:	commutative loops, construction of loops, matrices over finite fields, quadratic extensions [PDF]

	with Denis Simon	Journal of Algebra and its Applications 14,3 (2014), 20 pages
Abstract:	We study a construction introduced by Aleš Drápal, giving raise to commutative A-loops of order kn where k and n are odd numbers. We show which combinations of k and n are possible if the construction is based on a field or on a cyclic group. We conclude that if p and q are odd primes, there exists a non-associative commutative A-loop of order pq if and only if p divides q² – 1 and such a loop is most probably unique.
Keywords:	commutative A-loops, constructions of loops, matrices over finite fields, eigenvalues, quadratic extensions [PDF]

	with Michael K. Kinyon and Petr Vojtěchovský	Communications in Algebra 38,9 (2010), 3243–3267
Abstract:	A loop whose inner mappings are automorphisms is an automorphic loop (or A-loop). We characterise commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterisation and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of small orders and also of order p³, where p is a prime.
Keywords:	commutative automorphic loop, commutative A-loop, automorphic inner mappings, enumeration of A-loops [PDF]

	with Michael K. Kinyon and Petr Vojtěchovský	Transactions of American Mathematical Society 363,1 (2011), 365–384
Abstract:	An automorphic loop or an A-loop is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and (xy)^-1=x^-1y^-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. For each set π of primes, Q has a Hall π-subloop. If there is a finite simple nonassociative commutative A-loop, it is of exponent 2.
Keywords:	commutative automorphic loop, commutative A-loop, automorphic inner mappings, structure of loops [PDF]

	with Aleš Drápal	European Journal of Combinatorics, 31,7 (2010), 1907–1923
Abstract:	We start by describing all the varieties of loops Q that can be defined by autotopisms α_x, x∈Q, where α_x is a composition of two triples, each of which becomes an autotopism when the element x belongs to one of the nuclei. In this way we obtain a unifying approach to Bol, Moufang, extra, Buchsteiner and conjugacy closed loops. We reprove some classical facts in a new way and show how Buchsteiner loops fit into the traditional context. In 6 we describe a new class of loops with coincinding left and right nuclei. These loops have remarkable properties and do not belong to any of the classical classes.
Keywords:	loops, nucleus [PDF]

	with David Stanovský	Algebra Universalis 82,3 (2021)
Abstract:	We are interested in abstract conditions that characterize homomorphic images of affine quandles. Our main result is a two-fold characterization of this class: one by a property of the displacement group, the other one by a property of the corresponding affine mesh. As a consequence, we obtain efficient algorithms for recognizing homomorphic images of affine quandles, including an efficient explicit construction of the covering affine quandle.
Keywords:	Quandles, medial quandles, abelian quandles, quandle quotients [PDF]

		Publicationes Mathematicae 95,3–4 (2019), 505–514
	Abstract:	We present a construction of a family of involutory latin quandles, a family that contains all non-Alexander involutory latin quandles of order pq, for p<q odd primes. Such quandles exist if and only if p divides q²−1.
	Keywords:	Involutory quandles, quasigroups, Bruck loops, finite fields [PDF]

	with Agata Pilitowska, David Stanovský and Anna Zamojska-Dzienio	J. Algebra 510,15 (2018) 259–288
Abstract:	A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming from the commutator theory, and by an explicit construction over abelian groups. As a consequence, we obtain efficient algorithms for recognizing affine and quasi-affine quandles, and we enumerate small quasi-affine quandles. We also prove that the "abelian implies quasi-affine" problem of universal algebra has affirmative answer for the class of quandles.
Keywords:	Quandles, medial quandles, affine quandles, commutator theory, abelian algebras, quasiaffine algebras, quasi-affine modes [PDF]

	with David Stanovský and Petr Vojtěchovský	Discrete Mathematics 340,3 (2017) 404–415
Abstract:	We enumerate three classes of non-medial quasigroups of order 243 = 3⁵ up to isomorphism. There are 92 non-medial distributive quasigroups of order 243 (extending the work of Kepka and Němec), 17004 non-medial trimedial quasigroups of order 243 (extending the work of Kepka, Bénéteau and Lacaze), 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, automorphism groups of commutative Moufang loops, and computer calculations with the LOOPS package in GAP.
Keywords:	Distributive quasigroup, trimedial quasigroup, medial quasigroup, entropic quasigroup, commutative Moufang loop, latin square, Mendelsohn triple system, classification, enumeration.[PDF]

	with Agata Pilitowska and Anna Zamojska-Dzienio	Algebra Universalis 78,1 (2017), 43–54
Abstract:	This paper brings the construction of free medial quandles as well as free n-symmetric medial quandles and free m-reductive medial quandles.
Keywords:	Quandles, medial quandles, binary modes, free algebras [PDF]

	with Agata Pilitowska and Anna Zamojska-Dzienio	Communications in Algebra 46,11 (2018), 4803–4829
Abstract:	We classify subdirectly irreducible medial quandles. We show that in the finite case they are either connected (and therefore affine) or reductive. Moreover, we give an explicit description of all subdirectly irreducible reductive medial quandles.
Keywords:	Quandles, medial quandles, binary modes, reductive algebras, subdirectly irreducible algebras [PDF]

	with Agata Pilitowska, David Stanovský and Anna Zamojska-Dzienio	Journal of Algebra 43,1 (2015) 300–334
Abstract:	Medial quandles are represented using a heterogeneous affine structure. As a consequence, we obtain numerous structural properties, including enumeration of isomorphism classes of medial quandles up to 13 elements.
Keywords:	Quandles, medial quandles, groupoid modes, SIE groupoids, differential groupoids, reductive groupoids, medial idempotent groupoids, enumeration of quandles [PDF]

		Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 51,2 (2012) 67–72
	Abstract:	We investigate the following classes of the left distributive groupoids: left distributive left quasigroups, left divisible left distributive groupoids and left cancellative left idempotent left distributive groupoids and we conjecture that they are described by the same equational theories. More precisely, we translate the problem into the existence problem of a given groupoid.
	Keywords:	left distributivity, left idempotency, variety [PDF]

		Mathematica Slovaca 60,2 (2010), 213–222
	Abstract:	We study here so called cuts of terms and their classes modulo the identities of the left distributivity and the idempotency. We give an inductive definition of such classes and this gives us a criterion that might decide whether two terms are equivalent modulo both identities
	Keywords:	word problem, left distributivity, idempotency [PDF]

		Algebra and Discrete Mathematics, no. 4 (2006), 12–39
	Abstract:	We construct here the geometry monoids of LDI (left distributive idempotent) and of LDLI (left distributive left idempotent) identities. We study their properties and construct a monoid with solvable word problem based on relations of the geometry monoid of LDLI.
	Keywords:	geometry monoids, left distributivity, idempotency, word problem [PDF]

		Commentationes Mathematicae Universitas Carolinae 46,1 (2005) 15–20
	Abstract:	We study the groupoids satisfying both the left distributivity and the left idempotency laws. We show that they possess a canonical congruence admitting an idempotent groupoid as factor. This congruence gives a construction of left idempotent left distributive groupoids from left distributive idempotent groupoids and right constant groupoids.
	Keywords:	groupoids, left distributivity, left idempotency [PDF]

	with Katarzyna Matczak and Anna Mućka	Algebra and Discrete Mathematics 27,1 (2019), 3594–3610
Abstract:	In 1995 D. V. Belkin described the lattice of quasivarieties of modules over principal ideal domains. Our paper provides a description of the lattice of subquasivarieties of the variety of modules over a given Dedekind ring. It also shows which subvarieties of these modules are deductive (a variety is deductive if every subquasivariety is a variety).
Keywords:	quasivarieties, lattices, modules, Dedekind rings [PDF]

		Algebra Universalis, 57,3 (2007), 259–272
	Abstract:	The semidirect product of lattices is a lattice analogue of the semidirect product of groups. In this article we introduce this construction, show some basic facts and study a class of lattices closed under semidirect products. We also generalize this notion presenting the semidirect product of semilattices.
	Keywords:	lattices, semidirect product, semilattices [PDF]

		Communications in Algebra, 33,5 (2005) 1447–1460
	Abstract:	Let W be a (finite or infinite) Coxeter group and W_X be a proper standard parabolic subgroup of W. We show that the semilattice made up by W equipped with the weak order is a semidirect product of two smaller semilattices associated with W_X.
	Keywords:	Coxeter groups, weak order, lattice, semidirect product [PDF]

	with Agata Pilitowska	Submitted
Abstract:	We give a complete characterization of all indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2. In the first step we present a construction of some family of such solutions and in the second step we prove that every indecomposable involutive solution of the Yang-Baxter equation with multipermutation level 2 is a homomorphic image of a solution previously constructed. Analyzing this epimorphism, we are able to obtain all such solutions up to isomorphism and enumerate these of small sizes.
Keywords:	Yang-Baxter equation, set-theoretic solution, multipermutation solution, indecomposable solution, permutation group [PDF]

	with Marco Bonatto	to appear in Journal of Algebra and its Applications
Abstract:	Skew braces are algebraic structures related to the solutions of the set-theoretic quantum Yang- Baxter equation. We develop the central nilpotency theory for such algebraic structures in the sense of Freese-McKenzie [12] and we compare the universal algebraic notion of central nilpotency with the notion of right and left nilpotency developed in [7].
Keywords:	center, nilpotency, skew braces[PDF]

	with Agata Pilitowska and Anna Zamojska-Dzienio	Proceedings of the American Mathematical Society 150,10 (2022) 4223–4239
Abstract:	We study indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with cyclic permutation groups (cocyclic solutions). We give a complete system of three invariants for finite non-isomorphic solutions of this type and use this construction to enumerate all of them.
Keywords:	cocyclic brace, indecomposable set-theoretic solution, Yang-Baxter equation[PDF]

	with Agata Pilitowska and Anna Zamojska-Dzienio	Forum mathematicum 33,5 (2021) 1083–1096
Abstract:	We present a construction of all finite indecomposable involutive solutions of the Yang-Baxter equation of multipermutational level at most 2 with abelian permutation group. As a consequence, we obtain a formula for the number of such solutions with a fixed number of elements. We also describe some properties of the automorphism groups in this case - in particular, we show they are regular abelian groups.
Keywords:	Yang-Baxter equation, set-theoretic solution, multipermutation solution, transitive group [PDF]

	with Agata Pilitowska and Anna Zamojska-Dzienio	International Journal of Algebra and Computation 30,3 (2020) 667-683
Abstract:	We investigate a class of non-involutive solutions of the Yang-Baxter equation which generalize derived (self-distributive) solutions. In particular, we study generalized multipermutation solutions in this class. We show that the Yang-Baxter (permutation) groups of such solutions are nilpotent. We formulate the results in the language of biracks.
Keywords:	Yang-Baxter equation, set-theoretic solution, multipermutation solution, one-sided quasigroup, birack, left distributivity, rack, brace [PDF]

	with Agata Pilitowska and Anna Zamojska-Dzienio	Journal of Combinatorial Theory. Series A 176 (2020)
Abstract:	We study involutive set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2. These solutions happen to fall into two classes -- distributive ones and non-distributive ones. The distributive ones can be effectively constructed using a set of abelian groups and a matrix of constants. Using this construction, we enumerate all distributive involutive solutions up to size 14. The non-distributive solutions can be also easily constructed, using a distributive solution and a permutation.
Keywords:	Yang-Baxter equation, set-theoretic solution, multipermutation solution, one-sided quasigroup, birack, left distributivity, rack [PDF]

with Agata Pilitowska and Anna Zamojska-Dzienio

Journal of Pure and Applied Algebra 223,8 (2019), 3594–3610

Abstract:

Etingof, Schedler and Soloviev introduced, for each non-degenerate involutive set-theoretical solution (X,σ,τ) of the Yang-Baxter equation, the equivalence relation ~ defined on the set X and they considered a new non-degenerate involutive induced retraction solution defined on the quotient set X^~.

It is well known that translating set-theoretical non-degenerate solutions of the Yang-Baxter equation into the universal algebra language we obtain an algebra called a birack. In the paper we introduce the generalized retraction relation ≈ on a birack, which is equal to ~ in an involutive case. We present a complete algebraic proof that the relation ≈ is a congruence of the birack. Thus we show that the retraction of a set-theoretical non-degenerate solution is well defined not only in the involutive case but also in the case of all non-involutive solutions.

Keywords:

Yang-Baxter equation, set-theoretical solution, retraction of a solution, one-sided quasigroup, birack, congruence of an algebra [PDF]

	with Josef Zeman	Trends in Agricultural Engineering 2013 : conference proceedings, 282–286
Abstract:	When measuring the maximal energy output from solar panels over the year, we obtain, as optimal, the orientation to the south since the yield from the direct sun outshines the influence of the dispersed light form other directions. But there can be situations, like e.g. insular systems, where we do not look for the maximal output during sunny days since in these cases the storage batteries simply recharge and we cannot benefit of the additional power. We want rather to glean the sunlight during the days when we hardly recognize where the sun is since in these days we also want to power the instruments. Using the data from the direction intensity sensors installed on the roof of the CULS rectorate we figured out that over four years of observation during different seasons the maximum of the noon energy does not come from the south but from a direction 11 degrees to the west. Although it is currently, from the energetic point of view, a speculation, we would like to present the results of our several-year data collection about the direction of the emission, in hope that someone would verify the discovery directly on solar panels.
Keywords:	dispersed sunlight, solar panels

		Acta Universitatis Carolinae, Mathematica et Physica 53,1 (2012), 73–75
	Abstract:	In this article we answer the following question: if one has a ring R of characteristics 2 satisfying x^p = x, for some p; which values of p imply the identity x² = x?
	Keywords:	boolean ring, unitary ring, characteristic 2 [PDF]

		Analele Științifice ale Universității Ovidius Constanța, 18,2 (2010) 125–130
	Abstract:	The first phase of the general number field sieve is the polynomial selection. One popular method of selecting the polynomial was described by Murphy. A step in Murphy's polynomial selection consists of finding a minimum of an integral. The size of the coefficients of the polynomial causes that the classical steepest descent algorithm is not too effective. This article brings an idea how to improve the steepest descent algorithm so that it converges better and faster.
	Keywords:	integer factorisation, GNFS, polynomial selection, steepest descent [PDF]

French title: Treillis, groupes de Coxeter et les systèmes LDI
Czech title: Svazy dělitelnosti v monoidech kladných pletenců

Abstract (french, english and czech)

The thesis is bilingual, even pages written in french, odd pages written in czech so that one can easily verify that both versions match. The thesis has been defended 24 july 2004 in Caen.

french pages [PDF],
czech pages [PS], [PDF],
short english version [PS], [PDF],
slides from the presentation [PS], [PDF].

Commutative automorphic loops
Defended 1^st November 2017 [PDF]

Publications of Přemysl Jedlička