ContentsIntroduction ??1. The concept of hyperidentity. Hyperidentities of binary representations1.1. The concept of hyperidentity: examples1.2. Binary Cayley theorems for semigroups, idempotent and commutative semigroups, groups and the multiplicative groups of fields1.3. Stochastic algebras. The concept of coidentity ??2. The category of algebras of the same arithmetic type2.1. Homomorphisms and automorphisms of algebras of the same arithmetic type2.2. Hypervarieties of algebras ??3. The category of systems of the same arithmetic type. Nonclassical semantics in second order language ??4. Non-trivial associative and distributive hyperidentities in algebras that are invertible or close to being invertible Bibliography

Hyperidentities in algebras and varieties

We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone ℛℛ of distance matrices, and consider geometric and probabilistic problems connected with this object. The notion of the universal distance matrix is defined and we proved that the set of such matrices is everywhere dense G δ set in weak topology in the cone ℛ. Universality of distance matrix is the necessary and sufficient condition on the distance matrix of the countable everywhere dense set of so called universal Urysohn space which he had defined in 1924 in his last paper. This means that Urysohn space is generic in the set of all Polish spaces. Then we consider metric spaces with measures (metric triples) and define a complete invariant: its — matrix distribution. We give an intrinsic characterization of the set of matrix distributions, and using the ergodic theorem, give a new proof of Gromov’s “reconstruction theorem”. A natural construction of a wide class of measures on the cone ℛ is given and for these we show that with probability one a random Polish space is again the Urysohn space. There is a close connection between these questions, metric classification of measurable functions of several arguments, and classification of the actions of the infinite symmetric group ([15, 16]).

https://link.springer.com/content/pdf/10.1007%2F978-1-4020-2469-6_6.pdf

Random and Universal Metric Spaces

We give a review of some known published applications of quasi- groups in cryptology. Keywords: cryptology, quasigroup, (r,s,t)-quasigroup, stream- cipher, secret-sharing system, zero knowledge protocol, authen- tication of a message, NLPN sequence, Hamming distance.

/pdf/quasigroups-in-cryptology-80neg8jkg0.pdf

Quasigroups in cryptology

The notion of binary representation of algebras with at most two binary operations is introduced in this paper, and the binary version of Cayley theorem for distributive lattices is given by hyperidentities. In particular, we get the binary version of Cayley theorem for DeMorgan and Boolean algebras.

Binary representations of algebras with at most two binary operations. a cayley theorem for distributive lattices

Review: Gr. C. Moisil, Recherches sur l'algebre de la logique

A characterisation is given of the algebras with hyperidentities of the variety of Boolean algebras. A structure theorem is proven for such algebras. As a preliminary the hyperidentities of the varieties of lattices, modular lattices, and distributive lattices are characterised: existence theorems are proved for finite bases of hyperidentities.

Algebras with hyperidentities of the variety of Boolean algebras

In this article hyperidentities of the variety of Boolean algebras are characterized. A corollary of this characterization is the determination of a finite basis of hyperidentities for this variety. However, the class of polynomial algebras of Boolean algebras does not have a finite basis of hyperidentities.

Hyperidentities of Boolean Algebras

The hyperidentities of the variety of De Morgan algebras are characterized in this paper. A finite base of hyperidentities for this variety is found as a consequence. In particular, we obtain that the variety of De Morgan algebras has a decidable hyperequational theory. And finally we prove that the hyperequational theory of this variety is not one-based.

Yu. M. Movsisyan

Papers

Hyperidentities in algebras and varieties

Binary representations of algebras with at most two binary operations. a cayley theorem for distributive lattices

Algebras with hyperidentities of the variety of Boolean algebras

Hyperidentities of Boolean Algebras

Hyperidentities of De Morgan algebras