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Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : (N μ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q1 and Q2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form (α , α , γ) . [ABSTRACT FROM AUTHOR]

We initiate a systematic study of abelian-by-cyclic Moufang loops, that is, Moufang loops Q with an abelian normal subgroup X such that Q / X is a cyclic group. Among other results, we construct all split abelian-by-cyclic Moufang loops in which both X and Q / X are 3-divisible, using so-called Moufang permutations on X, which are permutations that deviate from an automorphism of X by an alternating biadditive mapping. Additional abelian-by-cyclic Moufang loops are obtained from so-called construction pairs. As an aside, we show that, in a Moufang loop Q obtained from a construction pair, the abelian normal subgroup X induces an abelian congruence of Q if and only if Q is a group. [ABSTRACT FROM AUTHOR]

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 $. [ABSTRACT FROM AUTHOR]

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 χ is the extended quadratic character on $\mathbb F$. Let $Q_{a,b}$ be the quasigroup over $\mathbb F$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi(y-x) \geqslant 0$ , 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 $\alpha\in \operatorname{Aut}(\mathbb F)$. We also characterize $\operatorname{Aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric. In proving our results, we also characterize the minimal subquasigroups of $Q_{a,b}$. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 (x u) (v y) = (x v) (u y) , (x y) (y x) = y and x x = 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The main concern of this paper are quasigroups of order nine that possess at most 18 associative triples. The order nine is the least order for which there exists a quasigroup (Q,*) such that x * (y * z)=(x * y) * z holds if and only if x=y=z. Up to isomorphism there is only one such quasigroup of this order. It has remarkable properties that bind it to a nearfield, to a PMD (9,4) and to a Sudoku division square. [ABSTRACT FROM AUTHOR]

Let Q be a quasigroup. Put a(Q)=∣{(x,y,z)∈Q3;x(yz)=(xy)z}∣ and assume that ∣Q∣=n. Let δL and δR be the number of left and right translations of Q that are fixed point free. Put δ(Q)=δL+δR. Denote by i(Q) the number of idempotents of Q. It is shown that a(Q)≥2n−i(Q)+δ(Q). Call Q extremely nonassociative if a(Q)=2n−i(Q). The paper reports what seems to be the first known example of such a quasigroup, with n=8, a(Q)=16, and i(Q)=0. It also provides supporting theory for a search that verified a(Q)≥16 for all quasigroups of order 8. [ABSTRACT FROM AUTHOR]

A directed triple system can be defined as a decomposition of a complete digraph to directed triples 〈x, y, z〉. By setting xy = z, yz = x, xz = y and uu = u we get a binary operation that can be a quasigroup. We give an algebraic description of such quasigroups, explain how they can be associated with triangulated pseudosurfaces and report enumeration results. [ABSTRACT FROM AUTHOR]

Let Q be a Buchsteiner loop. We describe the associator calculus in three variables, and show that |Q| ≥ 32 if Q is not conjugacy closed. We also show that |Q| ≥ 64 if there exists x ∈ Q such that x2 is not in the nucleus of Q. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that has both nilpotency class 3 and an abelian inner mapping group. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that given a Steiner triple system (STS) one can define a binary operation upon its base set by assigning for all and , where is the third point in the block containing the pair . The same can be done for Mendelsohn triple systems (MTSs) as well as hybrid triple systems (HTSs), where is considered to be ordered. In the case of STSs and MTSs, the operation is a quasigroup, however this is not necessarily the case for HTSs. In this paper we study the binary operation induced by HTSs. It turns out that each such operation satisfies for all and from the base set. We call every binary operation that fulfils this condition hybridly symmetric. Not all idempotent hybridly symmetric operations can be obtained from HTSs. We show that these operations correspond to decompositions of a complete digraph into certain digraphs on three vertices. However, an idempotent hybridly symmetric quasigroup always comes from an HTS. The corresponding HTS is then called a latin HTS (LHTS). The core of this paper is the characterization of LHTSs and the description of their existence spectrum. [Copyright &y& Elsevier]

Abstract: We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. We computationally prove Tutte’s conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20. [Copyright &y& Elsevier]

After exhibiting basic properties of Buchsteiner loops, we introduce a construction based on rings that produces a Buchsteiner loop of order 128 that is not conjugacy closed (CC). We also show that all Buchsteiner loops of nilpotency class two are CC and discuss the squaring mapping of those CC and Buchsteiner loops Q in which both Q/N and A ≤ Z(Q) are boolean groups (N is the nucleus and A is the associator subloop). We prove that this mapping determines Q up to isomorphism when Q/A, N, and the action of Q/A upon N are known. [ABSTRACT FROM AUTHOR]

Buchsteiner loops are those which satisfy the identity x \ (xy · z) = (y · zx) / x. We show that a Buchsteiner loop modulo its nucleus is an abelian group of exponent four, and construct an example where the factor achieves this exponent. [ABSTRACT FROM AUTHOR]