The Generalized Elliptic Curves $(GECs)$ are pairs $(Q,T)$, where $T$ is a family of triples $(x,y,z)$ of “points” from the set $Q$ characterized by equalities of the form $x.y=z$, where the law $x.y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x*y=u.(x.y)$. When $(x.y).(a.b)=(x.a).(y.b)$, identically $(Q,T)$ is an entropic $GEC$ and $(Q,*)$ is an abelian group. Similarly, a terentropic $GEC$ may be characterized by $x^2.(a.b)=(x.a)(x.b)$ and $(Q,*)$ is then a Commutative Moufang Loop $(CML)$. If in addition $x^2=x$, we have Hall $GECs$ and $(Q,*)$ is an exponent $3$

Term substitution induces an associative operation on the free objects of any equational variety. In the case of left distributivity, the construction can be extended to any monogenic structure.

We discuss a concept of loopoid as a non-associative generalization of Brandt groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.

In this paper, we define and study the hyper S-posets over an ordered semihypergroup in detail. We introduce the hyper version of a pseudoorder in a hyper S-poset, and give some related properties. In particular, we characterize the structure of factor hyper S-posets by pseudoorders. Furthermore, we introduce the concepts of order-congruences and strong order-congruences on a hyper S-poset A, and obtain the relationship between strong order-congruences and pseudoorders on A. We also characterize...

Given a groupoid $\langle G,☆\rangle$, and $k\ge 3$, we say that $G$ is antiassociative if an only if for all $x_1,x_2,x_3\in G$, $(x_1☆x_2)☆x_3$ and $x_1☆(x_2☆x_3)$ are never equal. Generalizing this, $\langle G,☆\rangle$ is $k$-antiassociative if and only if for all $x_1,x_2,...,x_k\in G$, any two distinct expressions made by putting parentheses in $x_1☆x_2☆x_3☆\cdots ☆x_k$ are never equal. We prove that for every $k\ge 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.

It is well known that given a Steiner triple system one can define a quasigroup operation $·$ upon its base set by assigning $x·x=x$ for all $x$ and $x·y=z$, where $z$ is the third point in the block containing the pair $\{x,y\}$. The same can be done for Mendelsohn triple systems, where $(x,y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed triple systems....

Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,\circ )$ which arises as a result of a construction in “Engelsche elemente noetherscher gruppen” (1957) by R. Baer. We investigate some general properties and applications of “$\circ$” and determine a necessary and sufficient condition on $G$ in order for $(G,\circ )$ to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that $G$ is metabelian if and only if $(G,\circ )$ is an automorphic loop. We...

Quasigroups were originally described combinatorially, in terms of existence and uniqueness conditions on the solutions to certain equations. Evans introduced a universal-algebraic characterization, as algebras with three binary operations satisfying four identities. Now, quasigroups are redefined as heterogeneous algebras, satisfying just two conditions respectively known as hypercommutativity and hypercancellativity.

We give new equations that axiomatize the variety of trimedial quasigroups. We also improve a standard characterization by showing that right semimedial, left F-quasigroups are trimedial.