We construct non-associative key establishment protocols for all LD-, multi-LD-, and other left-distributive systems. 2010 Mathematics Subject Classification. 20N02, 20F36.

In this work, two normal congruences are built on two quasigroups with underlining set Zn relative to the linear-bivariate polynomial P (x, y) = a+ bx+ cy that generates a quasigroup over the ring Zn. Four quasigroups are built using the normal congruences and these are shown to be homomorphic to the quasigroups with underlining set Zn. Some subquasigroups of the quasigroups with underlining se...

We apply a general approach for distributions of binary isolating and semi-isolating formulas to families of isolated types and to the class of countably categorical theories. Containing compositions and Boolean combinations, structures associated with binary formulas linking isolated types in general case and for ω-categorical theories are characterized. Mathematics Subject Classification: 03C...

Generalized hypersubstitutions were introduced by S. Leeratanavalee and K. Denecke as a way of making precise the concepts of strong hyperidentity and M strong hyperidentity. A variety V is called M strongly solid if every identity of V is an M strong hyperidentity. We produce a monoid M for which the k-normalized variety Nv k (V ) of an N -strongly solid variety V with respect to a valuation v...

The study of associativity of the commutator operation in groups goes back to the work of F. W. Levi in 1942. In the 1960’s Richard J. Thompson created a group F whose elements are representatives of the generalized associative law for an arbitrary binary operation. In 2006, Geoghegan and Guzmán proved that a group G is solvable if and only if the commutator operation in G eventually satisfies ...

By left magma-$e$-magma, I mean a set containingthe fixed element $e$, and equipped by two binary operations "$cdot$", $odot$ with the property $eodot (xcdot y)=eodot(xodot y)$, namelyleft $e$-join law. So, $(X,cdot,e,odot)$ is a left magma-$e$-magmaif and only if $(X,cdot)$, $(X,odot)$ are magmas (groupoids), $ein X$ and the left $e$-join law holds.Right (and two-sided) magma-$e$-magmas are de...

Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthoimplication algebras. These are in a natural one-to-one correspondence with families of compatible orthomodular lattices. It is proved that congruence kernels of orthoimplication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the correspondin...

Laver tables An are certain finite shelves (i.e., sets endowed with a binary operation distributive with respect to itself). They originate from Set Theory and, in spite of an elementary definition, have complicated combinatorial properties. They are conjectured to approximate the free monogenerated shelf F1, this conjecture being currently proved only under a large cardinal axiom. This talk is...

We complete Artin’s braid group B∞ with some limit points (with respect to a natural topology), thus obtaining an extended monoid where new left self-distributive operations are defined. This construction provides an effective realization for some free algebraic system involving a left distributive operation and a compatible associative product. AMS Subject classification: 20F36, 20N02 Here we ...

Given a group, there is a natural operation of conjugation, defined by x ∗ y = xyx−1. We study the variety generated by all G(∗), G a group. In particular, we are concerned about the question, whether this variety has finitely based equational theory. We study the equations satisfied by group conjugation. Strictly speaking, for a group G we put x∗y = xyx−1 for all x, y ∈ G and we ask, which equ...