In a groupoid, consider arbitrarily parenthesized expressions on the $k$ variables $x_0,x_1,\cdots x_{k-1}$ where each $x_i$ appears once and all variables appear in order of their indices. We call these expressions $k$-ary formal products, and denote the set containing all of them by $F^{\sigma }\left(k\right)$. If $u,v\in F^{\sigma }\left(k\right)$ are distinct, the statement that $u$ and $v$ are equal for all values of $x_0,x_1,\cdots x_{k-1}$ is a generalized associative law. Among other results, we show that many small groupoids are completely dissociative, meaning that no generalized associative law holds...

A ternary ring is an algebraic structure $ℛ=(R;t,0,1)$ of type $(3,0,0)$ satisfying the identities $t(0,x,y)=y=t(x,0,y)$ and $t(1,x,0)=x=(x,1,0)$ where, moreover, for any $a$, $b$, $c\in R$ there exists a unique $d\in R$ with $t(a,b,d)=c$. A congruence $\theta$ on $ℛ$ is called normal if $ℛ/\theta$ is a ternary ring again. We describe basic properties of the lattice of all normal congruences on $ℛ$ and establish connections between ideals (introduced earlier by the third author) and congruence kernels.

In this short note, it is shown that if $A,B$ are $H$-connected transversals for a finite subgroup $H$ of an infinite group $G$ such that the index of $H$ in $G$ is at least 3 and $H\cap H^u\cap H^v=1$ whenever $u,v\in G\setminus H$ and $u{v}^{-1}\in G\setminus H$ then $A=B$ is a normal abelian subgroup of $G$.

Let $q$ be a positive integer. An algebra is said to have the property $(2,q)$ if all of its subalgebras generated by two distinct elements have exactly $q$ elements. A variety $𝒱$ of algebras is a variety with the property $(2,q)$ if every member of $𝒱$ has the property $(2,q)$. Such varieties exist only in the case of $q$ prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property $(2,q)$, $2<q$, blocks of Steiner system of type $(2,q)$ are obtained. The stated correspondence between Steiner...

We give a short account of the construction and properties of left neofields. Most useful in practice seem to be neofields based on the cyclic group and particularly those having an additional divisibility property, called D-neofields. We shall give examples of applications to the construction of orthogonal latin squares, to the design of tournaments balanced for residual effects and to cryptography.