Search

We study abelian-by-cyclic Moufang loops. We construct all split $3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups $(X,+)$, which are permutations that deviate from an automorphism of $(X,+)$ by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops $Q$ obtained from a construction pair on $(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of $Q$ if and only if $Q$ is a group.

We prove that a normal subloop $X$ of a Moufang loop $Q$ induces an abelian congruence of $Q$ if and only if each inner mapping of $Q$ restricts to an automorphism of $X$ and $u(xy) = (uy)x$ for all $x,y\in X$ and $u\in Q$. The former condition can be omitted when $X$ is $3$-divisible. This characterization is then used to show that classically solvable finite $3$-divisible Moufang loops are congruence solvable.

In groups, an abelian normal subgroup induces an abelian congruence. We construct a class of centrally nilpotent Moufang loops containing an abelian normal subloop that does not induce an abelian congruence. On the other hand, we prove that in $6$-divisible Moufang loops, every abelian normal subloop induces an abelian congruence. In loops, congruence solvability adopted from the universal-algebraic commutator theory of congruence modular varieties is strictly stronger than classical solvability adopted from group theory. It is an open problem whether the two notions of solvability coincide in Moufang loops. We prove that they coincide in $6$-divisible Moufang loops and in Moufang loops of odd order. In fact, we show that every Moufang loop of odd order is congruence solvable, thus strengthening Glauberman's Odd Order Theorem for Moufang loops., Minor revision. Includes a shorter proof of Theorem 1.11 for the case p=3. To appear in Journal of Algebra

Let $\mathbb{F}$ be a finite field of odd order and $a,b\in\mathbb{F}\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where $\chi$ is the quadratic character. Let $Q_{a,b}$ be the quasigroup upon $\mathbb{F}$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi(y-x) \ge 0$, and $(x,y)\mapsto x+b(y-x)$ if $\chi(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$ if and only if $\{a,b\}= \{\alpha(c),\alpha(d)\}$ for some $\alpha\in \textrm{aut}(\mathbb{F})$. We also characterise $\textrm{aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, medial, left or right distributive, flexible or semisymmetric.

Examination of thermal decomposition of street samples of cocaine and methamphetamine shows that typical products detected in previous studies are accompanied by a wide palette of simple volatile compounds easily detectable by spectral techniques. These molecules increase smoke toxicity and their spectral detection can be potentially used for identification of drug samples by well-controlled laboratory thermolysis in temperature progression. In our study, street samples of cocaine and methamphetamine have been thermolyzed under vacuum over the temperature range of 350–650 °C. The volatile products (CO, HCN, CH4, C2H4, etc.) have been monitored by high-resolution Fourier-transform infrared (FTIR) spectrometry in this temperature range. The decomposition mechanism has been additionally examined theoretically by quantum-chemical calculations for the highest temperature achieved experimentally in our study and beyond. Prior to analysis, the street samples have also been characterized by FTIR, Raman spectroscopy, energy-dispersive X-ray spectroscopy, and melting point determination.

Sentenced offenders who re-offend prior to serving their previously imposed sentence (multiple conviction offenders) are situated between multiple and repeat offenders. This article examines how they should be sentenced based on censure, consequentialist and desert theories. I conclude that these aims cannot be achieved if they are treated as repeat offenders, and neither can the requirement of proportionality. Censure is, similarly, communicated primarily via hard treatment not via sentence pronouncement. I further analyse all continental European penal codes; half of them do not have any provision governing the sentencing of multiple conviction offenders (tacitly treating them as repeat offenders) and only two countries provide detailed sentencing guidance. I conclude by offering recommendations for the principled sentencing of multiple conviction offenders.

Several varieties of quasigroups obtained from perfect Mendelsohn designs with block size 4 are defined. One of these is obtained from the so‐called directed standard construction and satisfies the law xy ⋅ (y ⋅ xy) = x and another satisfies Stein's third law xy ⋅ yx = y. Such quasigroups which satisfy the flexible law x ⋅ yx = xy ⋅ x are investigated and characterized. Quasigroups which satisfy both of the laws xy ⋅ (y ⋅ xy) = x and xy .yx = y are shown to exist. Enumeration results for perfect Mendelsohn designs PMD(9, 4) and PMD(12, 4) as well as for (nonperfect) Mendelsohn designs MD(8, 4) are given.

The assessment of cadmium content in the muscle, liver and kidneys of fattening pigs and sows was conducted in the Czech Republic during the period of 2015–2019. The samples were collected from a total of 32 sows and 210 fattening pigs. The mean cadmium content in the livers and kidneys of all sows was 0.102 ± 0.010 and 0.361 ± 0.032 mg·kg-1, respectively. The mean cadmium content in the livers and kidneys of all fattening pigs was 0.028 ± 0.001 and 0.141 ± 0.005 mg·kg-1, respectively. The maximum residual limit for human consumption was not exceeded in any sample. The cadmium content in the liver did not differ significantly between sows and fattening pigs in any particular year. In contrast, the cadmium content in kidneys differed significantly between sows and fattening pigs, being higher in sows (P < 0.05). No significant decreasing trend was observed for the cadmium content in tissues during the whole time period. It can be concluded that there is a need for further monitoring of the cadmium content in pig tissues in the Czech Republic.

The subject of this study is the susceptibility of the Czech Evangelical community to radicalization into violent extremism. The current socio-economic climate is difficult to handle for many, and the danger of violent extremism needs to be accounted for. With cases of right- wing, nationalist, and conservative attacks being well-known and studied, it is imperative to seek the understanding of communities, potentially susceptible to violent extremism. This ethnographic study, therefore, seeks to probe such susceptibility within the Czech Evangelical community. For this the study first introduces valuable information collected through research of radicalization, including the most relevant models used in the study of radicalization, and then uses qualitative semi-structured interviews conducted on ten members of said community, coupled with netnography and both passive and active observation, to reveal that the Czech Evangelical community is surprisingly resilient to radicalization into violent extremism yet endangered non the less, especially with regards to mental health of some of its members. At the same time, splinter groups and individuals should be viewed as the most critically endangered actors, as they lack the corrective influence, usually provided by the community.

Let the product of points A and B be the vertex C of the right isosceles triangle for which AB is the base, and ABC is oriented anticlockwise. This yields a quasigroup that satisfies laws (xu) (vy) = (xv)(uy), (xy)(yx) = y and xx = x. Such quasigroups are called quadratical. Quasigroups that satisfy only the latter two laws are equivalent to perfect Mendelsohn designs of length four (PMD(v,4)). This paper examines various algebraic identities induced by PMD(v,4), classifies finite quadratical quasigroups, and shows how the square structure of quadratical quasigroups is associated with toroidal grids.

A loop $X$ is said to satisfy Moufang's theorem if for every $x,y,z\in X$ such that $x(yz)=(xy)z$ the subloop generated by $x$, $y$, $z$ is a group. We prove that the variety $V$ of Steiner loops satisfying the identity $(xz)(((xy)z)(yz)) = ((xz)((xy)z))(yz)$ is not contained in the variety of Moufang loops, yet every loop in $V$ satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah.

Emission reduction has become an important subject of discussion in recent years, mostly due to CO2 emission limits set by the European Union to protect the environment. One of the most affected areas is the automotive industry for its usage of internal combustion engines. Even though it is still not possible to eliminate the use of fossil fuels completely, it can be decreased by increasing the efficiency of internal combustion engines. One of the ways to do so is using cylinder deactivation technology. This paper summarises the history of cylinder deactivation and describes various approaches to the implementation of this technology, including its potential and constraints. The synergy of cylinder deactivation with other technologies is presented and future possibilities are predicted.

Purpose Sentencing has been shown to be influenced by different extralegal factors. Following psychological research into the various effects the weather can have on mood and two studies claiming finding an influence of weather on sentencing, we examine the extent to which the weather may influence sentencing. Methods Using sentencing decisions from twelve district courts in Prague in the period 2011–2015 and multilevel modelling techniques, we explore the impact of temperature, wind speed, sunshine, precipitation, barometric pressure and humidity on the decision to incarcerate and the duration of non-suspended prison sentences. Results In line with the inconclusive findings in the psychological literature on weather and mood and contrary to previous two studies finding the link between weather and sentencing, we do not find that the weather has any substantial impact on sentencing decision making. Conclusions We conclude that no meaningful unwarranted disparities in sentencing are caused by the weather in Prague, Czech Republic.

Uncovering attacks on data and computer systems and those responsible for them is one of the contemporary problems that the authorities involved in criminal proceedings have to deal with. Where this sort of cybercrime is concerned we can expect not only high levels of latency but also a low clear-up rate for crimes on file. This paper demonstrates this using the example of the Czech Republic, by providing an analysis of all cases of this type of cybercrime dealt with by courts between 2008 and 2016. It focuses in particular on proceedings concerned with illegal access and data and system interference. The cybercrimes that are brought to court in the Czech Republic is not representative of the cybercrime committed in the Czech Republic either in terms of its type or its extent. The cases dealt with by the courts are primarily connected with personal relationships and the Internet. The most frequent type of case thus concerns the misuse of passwords for accessing social networking sites or e-mail inboxes. We were also surprised to find that the courts had begun to apply facts relevant to cybercrime cases in cases involving the manual manipulation of gambling machines. The Czech Republic has evidently not yet found a way of demonstrating criminals’ guilt in more complex cybercrime cases.

A division sudoku is a latin square whose all six conjugates are sudoku squares. We enumerate division sudokus up to a suitable equivalence, introduce powerful invariants of division sudokus, and also study latin squares that are division sudokus with respect to multiple partitions at the same time. We use nearfields and affine geometry to construct division sudokus of prime power rank that are rich in sudoku partitions.

Re-offending is usually operationalized as a binary variable, signaling whether the criminal justice system encountered a given offender within a specific time-period. An increase or decrease in the proportion of participants committing a certain type of offense is usually considered the primary measure of effectiveness of any intervention. Yet such a binary operationalization of re-offending does not provide a complete picture of re-offending and might provide distorted or even false results by failing to take into account the seriousness and frequency of the re-offending. This paper begins by outlining why re-offending should be measured in a non-binary way, discussing the aims of evaluated treatments, the black-and-white nature of dichotomization, results from studies on desistance, chronic offenders and the effectiveness of sanctions including incapacitation. I then discuss possible ways of measuring the seriousness and frequency of re-offending, with a specific focus on their measurement in the continental European context and on the measure of re-conviction (whereas re-arrest is usually used as the equivalent measure in common law countries). I discuss the possible use of sentencing ranges set by legislators or of starting points set within various forms of sentencing guidelines to measure seriousness, in a similar way as the Cambridge Crime Harm Index does. I pay particular attention to the operationalization of the re-conviction measure in studying the effect of sanctions.

Early Modern Perspectives on Islam in the Czech lands by Bartosz Paprocki and his Diadochus This thesis analyses views on Islam in the 16 and 17th century Czech Lands. Its perspective is not religionist, but rather based on confrontation with muslims (mostly Turks) as represented in selected work of Bartosz Paprocki. Specifically, the thesis looks for specific features in the authorʼs approach to the issue and reflects on them from the viewpoint of present-day historiography.

We study the implications of a structure of the criminal justice system on sentencing decisions. To limit sentencing disparities, offenses are typically divided into subsections with specific sentencing ranges. The classification into corresponding subsections often depends on exceeding a given quantity threshold, such as drug amount. We study the consequences of these quantity thresholds on sentencing decisions and argue that the threshold effect can be decomposed into two opposing mechanisms: the severity mechanism and the reference one. An experiment with Czech prosecutors shows that thresholds cause an enormous increase in harshness in sentencing, leading to sentencing disparities. We further introduce empirical measures of (in)justice and quantify the consequences of quantity thresholds on the probability of imposing a just sentence.

Sentencing in the Czech Republic: An Empirical Investigation Abstract Sentencing is in many respects still terra incognita. This holds true especially for other countries than common law ones and those in the Western and Northern Europe. This dissertation thus empirically studies sentencing practices in the Czech Republic using quantitative methods. It is composed of four articles focusing on various sentencing issues: Firstly, on the influence of judicial experience on sentencing using longitudinal data analyzing judges' trajectories from 2008 onward. It shows that increasing experience reduces inter-judge disparity. Secondly, on analyzing previously underappreciated measure to achieve policy change: Informal authority of the supreme institutions. In 2016 the Prosecutor General's Office and the Supreme Court organized several meetings and seminars for prosecutors and judges in order to persuade them to impose more fines. This paper illustrates that these informal tools might have been underappreciated as they are highly effective. The third paper is dedicated to studying a particular extra-legal characteristics influencing sentencing in Prague: The weather. It builds both on psychological literature showing that weather influences mood and mood often impacts behavior and decision-making and on previous US...

The assessment of mercury content in the muscle, liver and kidneys of fattening pigs and sows was conducted in the Czech Republic during the period of 2015–2019. The samples were collected from sows (n = 31) and fattening pigs (n = 210). The average mercury content in muscle, liver, and kidneys of sows over 5 years was 0.00057 ± 0.00004, 0.0045 ± 0.0008, and 0.0224 ± 0.0067 mg·kg-1, respectively. The average mercury content in muscle, liver, and kidneys of fattening pigs over the same period was 0.00059 ± 0.00004, 0.0012 ± 0.0001, and 0.0044 ± 0.0004 mg·kg-1, respectively. The maximum residual limit (MRL) for human consumption was exceeded in 14 kidney samples. The mercury content in liver did not differ significantly between sows and fattening pigs; in kidneys, the mercury content was higher in sows, with a significant difference in one year (P < 0.05). A slight decrease in the mercury content in kidneys and liver was observed in fattening pigs from the year 2015. In sows, there was a significant increase in the mercury content in kidneys in the year 2016. It can be concluded that there is a need for further monitoring of the mercury content in pig tissues in the Czech Republic.

Supplemental material, sj-pdf-1-euc-10.1177_1477370821996903 for Sentencing multiple conviction offenders by Jakub Drápal in European Journal of Criminology

Internal-combustion engines with a small number of cylinders can be difficult to test on a dynamometer due to the increased load of parts of the torsional chain between the engine and the dynamometer. This is due to the uneven rotation of the crankshaft, especially at low speeds. It is therefore advisable to perform appropriate simulations before the experiment itself. The paper describes a time-efficient simulation of the torsional dynamics of a two-cylinder engine for an electric vehicle range extender, connected via a complex torsional chain to an electric dynamometer for testing purposes. The simulations are performed in the frequency domain by means of an in-house software built in the Fortran programming language. Particular attention is paid to the disc clutch, which is significantly loaded in the case of this engine.

Let $Q$ be a loop. If $S\leq Q$ is such that $\varphi(S) \subseteq S$ for each standard generator of $\mathrm{Inn}(Q)$, then $S$ does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus. Loops that are both Buchsteiner and Osborn are characterized as loops in which each square is in the nucleus., Comment: 16 pages; v.2: typos fixed; bibliography entry fixed

Let q be an odd prime power and suppose that $a,b\in \mathbb {F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ if $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z \Leftrightarrow x=y=z$ . Denote by $\sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $\alpha \approx 0.029\,08$ and $\beta \approx 0.012\,59$ such that if $q\equiv 1 \bmod 4$ , then $\lim \sigma (q)/q^2 = \alpha $ , and if $q \equiv 3 \bmod 4$ , then $\lim \sigma (q)/q^2 = \beta $ .

We introduce the concept of propagating equations and focus on the case of associativity propagating in varieties of loops. An equation $\varepsilon$ propagates in an algebra $X$ if $\varepsilon(\overrightarrow y)$ holds whenever $\varepsilon(\overrightarrow x)$ holds and the elements of $\overrightarrow y$ are contained in the subalgebra of $X$ generated by $\overrightarrow x$. If $\varepsilon$ propagates in $X$ then it propagates in all subalgebras and products of $X$ but not necessarily in all homomorphic images of $X$. If $\mathcal V$ is a variety, the propagating core $\mathcal V_{[\varepsilon]} = \{X\in\mathcal V:\varepsilon$ propagates in $X\}$ is a quasivariety but not necessarily a variety. We prove by elementary means Goursat's Lemma for loops and describe all subdirect products of $X^k$ and all finitely generated loops in $\mathbf{HSP}(X)$ for a nonabelian simple loop $X$. If $\mathcal V$ is a variety of loops in which associativity propagates, $X$ is a finite loop in which associativity propagates and every subloop of $X$ is either nonabelian simple or contained in $\mathcal V$, then associativity propagates in $\mathbf{HSP}(X)\lor\mathcal V$. We study the propagating core $\mathcal S_{[x(yz)=(xy)z]}$ of Steiner loops with respect to associativity. While this is not a variety, we exhibit many varieties contained in $\mathcal S_{[x(yz)=(xy)z]}$, each providing a solution to Rajah's problem, i.e., a variety of loops not contained in Moufang loops in which Moufang Theorem holds., Comment: Minor updates to version 1

Foucaults zweite Vorlesungsreihe am College de France kreist um Anfange. Sie ist seine erste historische Studie, die sich dezidiert Praktiken des Strafens zuwendet, und lasst sich als Versuch verstehen, eine Geschichte der Gegenwart aus einer Nietzscheanischen Perspektive heraus zu schreiben. In der Niederschlagung des Aufstandes der Nu-Pieds im Jahre 1639 sieht Foucault ein „neues Repressionssystem“ in Erscheinung treten, aus dem die uns heute so gelaufigen Institutionen der Polizei, des Gefangnisses und des Gerichtswesens hervorgehen. Foucaults zweiter ‚Sprung‘ zuruck, diesmal in das germanische Recht des Mittelalters, erlaubt, die Geschichte ihrer Anfange zu schreiben und das Verhaltnis von Recht und Genealogie als Kritik neu zu reflektieren.

This paper presents the motivation for a range extender engine development because in comparison to power train of an electric vehicle equipped only by high-voltage battery, cost and mass saves should be reached depending on required vehicle range. The engine concept is supposed to be based on the cylinder unit of the standard ŠKODA AUTO three-cylinder engine. In order to meet power requirements and demands on costs and simplicity, the in-line two-cylinder layout is chosen for range extender. Six variants of the crank train layout are designed and compared considering mainly their balancing and vibration.

The paper presents tools for vibration evaluation of power train mounting system of a passenger car to meet demands for passenger’s comfort. The computational model is based on the Multi-Body System principles and includes submodules as are engine, clutch, gearbox, final drive, drive shafts and rheological models of all power train mountings. Simulation results are validated on account of measurement of power train vibration at different engine load conditions. Both simulation and measurement are carried out at power train of the European B-segment car with an in-line three-cylinder spark-ignition engine and manual gearbox. Presented methodology can bring time and cost savings particularly during mountings design process.

Although there is an extensive literature on sentencing disparities in common law countries, there have been only a few empirical studies in continental countries and virtually none in post-communist ones. This article presents findings from the Czech Republic, which show that there are important disparities in post-communist Europe, comparable to those in the USA before the introduction of sentencing guidelines. I employ a multi-level modelling approach to study the sentencing practices in Czech district courts for the three most common offences. The values of intra-cluster correlation are found to be between .066 and .178 for the various models, which is considered high. The specifics of civil law post-communist countries are further discussed in relation to choosing appropriate ways of reducing disparities.

The paper deals with research of an engine block load caused particularly by inertia effects of the crank train rotating parts. Substantial reduction of this load can be achieved by an appropriate design of the crankshaft counterweights. The relation between crank train internal moments of rotating parts and the engine block load is investigated in the case of an I4 MPI engine with two variants of the crank train design. Used methodology is based on MBS simulations, where modally reduced flexible bodies and computational model of a hydrodynamic bearing are included.

Penal populism has repeatedly been described as influencing penal policies, with harsh penal practices presented as evidence of its influence. However, little attention has yet been paid to its role in the development of penal policies in post-authoritarian countries, which generally have large prison populations. Some minimal research has suggested that Central European countries were driven by penal nationalism following the 1989 revolutions. I examine this claim for the Czech Republic, using Garland (2013) 's framework of the five dimensions of a penal state. My analysis of political manifestoes shows that Czech politicians did not employ “law and order” rhetoric. The country's penal reforms were led by lenient penal elites. Nevertheless, a lack of analysis, coordination and sufficient funding resulted in a failure to properly identify or tackle the causes of the country's high imprisonment rate. Even though it gradually became more difficult to impose prison sentences, insufficient attention was paid to the length of the sentences Czech prisoners were serving. The large Czech prison population thus seems to be the result of state actors’ negligence, but not of penal populism nor of penal nationalism.

Day fines are an expression of equality for rich and poor alike. Although the theoretical concept of day fines (also called unit or structured fines) has been thoroughly studied, there has not been any empirical study examining the link between the amount of one day fine and the wealth of an offender, to establish whether or not day fines in reality create equality between offenders with different levels of wealth. This paper reports a study conducted in the Czech Republic, which suggests that the concept of day fines does not function without a clear formula specifying how day fines should be calculated. A comparison of all states in Europe that use day fines indicates that only two states have such a formula. The other 20 states have no formula, and thus their propensity for misusing day fines may be high. Several other characteristics of day fines, such as the number of day fines issued and the conversion rate at which unpaid fines are transformed into days in prison, are also discussed.

High level of mechanical efficiency is exacted from internal-combustion engines. The reduction of friction losses of crankshaft main bearings can significantly contribute to the enhancement of this efficiency. For this purpose, an innovative design of a crankshaft is developed. The potential of computational modelling during the development of this innovative crank train is described in the article. The dynamics of the whole crank train is solved by using a multi-body system software, where flexible finite-element bodies along with hydrodynamic bearings are incorporated. Regarding the simulation results, attention is paid to the torsional vibration and its analysis, including concept design of a torsional damper, because a reduction of friction losses is associated with the improvement of torsional vibration in this case.

This paper presents a strategy for computational modelling of elastic rubber couplings under dynamic loading. Methods how to determine static and dynamic characteristics of the elastic coupling based on static and dynamic experimental tests of rubber elements are presented. The nonlinear deformation behaviour, frequency and temperature dependent properties of rubber are considered for computational models. The model is applied to the elastic coupling connecting an in-line six-cylinder natural gas engine and an electrical generator. Loading forces are based on in-cylinder pressure measurement. Experimental verification of the computational model results is carried out by measuring the values on a test engine using the non-contact laser measuring technique.

Let Q be a quasigroup. For $$\alpha ,\beta \in S_Q$$ let $$Q_{\alpha ,\beta }$$ be the principal isotope $$x*y = \alpha (x)\beta (y)$$ . Put $$\mathbf a(Q)= |\{(x,y,z)\in Q^3;$$ $$x(yz)) = (xy)z\}|$$ and assume that $$|Q|=n$$ . Then $$\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})$$ , and for every $$\alpha \in S_Q$$ there is $$\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2$$ , where $$f_x=|\{y\in Q;$$ $$ y = \alpha (y)x\}|$$ . If G is a group and $$\alpha $$ is an orthomorphism, then $$\mathbf a(G_{\alpha ,\beta })=n^2$$ for every $$\beta \in S_Q$$ . A detailed case study of $$\mathbf a(G_{\alpha ,\beta })$$ is made for the situation when $$G = \mathbb Z_{2d}$$ , and both $$\alpha $$ and $$\beta $$ are “natural” near-orthomorphisms. Asymptotically, $$\mathbf a(G_{\alpha ,\beta })>3n$$ if G is an abelian group of order n. Computational results: $$\mathbf a(7) = 17$$ and $$\mathbf a(8) \le 21$$ , where $$\mathbf a(n) = \min \{\mathbf a(Q);$$ $$ |Q|=n\}$$ . There are also determined minimum values for $$\mathbf a(G_{\alpha ,\beta })$$ , G a group of order $$\le 8$$ .

An equivalence $$\sim $$ upon a loop is said to be multiplicative if it satisfies $$x\sim y$$ , $$u \sim v \Rightarrow xu \sim yv$$ . Let X be a set with elements $$x\ne y$$ and let $$\sim $$ be the least multiplicative equivalence upon a free loop F(X) for which $$x\sim y$$ . If $$a,b\in F(X)$$ are such that $$a\ne b$$ and $$a\sim b$$ , then neither $$a\backslash c \sim b\backslash c$$ nor $$c/a \sim c/b$$ is true, for every $$c\in F(X)$$ .

A quasigroup $Q$ is called maximally nonassociative if for $x,y,z\in Q$ we have that $x\cdot (y\cdot z) = (x\cdot y)\cdot z$ only if $x=y=z$. We show that, with finitely many exceptions, there exists a maximally nonassociative quasigroup of order $n$ whenever $n$ is not of the form $n=2p_1$ or $n=2p_1p_2$ for primes $p_1,p_2$ with $p_1\le p_2

Gagola’s construction of cyclic extensions of Moufang loops for orders coprime to three is investigated with the goal of describing necessary and sufficient conditions when the construction really gives a Moufang loop. This is achieved both in terms of equations and loop autotopisms. The obtained description is then specialized to Moufang loops that cyclically extend a normal subgroup G, showing the connection to Chein’s construction in case when the semiautomorphism characterizing the extension is an antiautomorphism of G. These results are then illustrated upon the case when G is a noncommutative group that possesses a cyclic subgroup of index two and has a small center (i.e.|Z(G)|∈{1,2}).

We say that ( x , y , z ) ∈ Q 3 is an associative triple in a quasigroup Q ( ⁎ ) if ( x ⁎ y ) ⁎ z = x ⁎ ( y ⁎ z ) . It is easy to show that the number of associative triples in Q is at least | Q | , and it was conjectured that quasigroups with exactly | Q | associative triples do not exist when | Q | > 1 . We refute this conjecture by proving the existence of quasigroups with exactly | Q | associative triples for a wide range of values | Q | . Our main tools are quadratic Dickson nearfields and the Weil bound on quadratic character sums.