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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]

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]

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]

Let Q be a quasigroup. For α,β∈SQ let Qα,β be the principal isotope x∗y=α(x)β(y). Put a(Q)=|{(x,y,z)∈Q3;x(yz))=(xy)z}| and assume that |Q|=n. Then ∑α,βa(Qα,β)/(n!)2=n2(1+(n-1)-1), and for every α∈SQ there is ∑βa(Qα,β)/n!=n(n-1)-1∑x(fx2-2fx+n)≥n2, where fx=|{y∈Q;y=α(y)x}|. If G is a group and α is an orthomorphism, then a(Gα,β)=n2 for every β∈SQ. A detailed case study of a(Gα,β) is made for the situation when G=Z2d, and both α and β are “natural” near-orthomorphisms. Asymptotically, a(Gα,β)>3n if G is an abelian group of order n. Computational results: a(7)=17 and a(8)≤21, where a(n)=min{a(Q);|Q|=n}. There are also determined minimum values for a(Gα,β), G a group 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]

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]

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. [ABSTRACT FROM AUTHOR]