Semivarieties and Schreier varieties of unary algebras

A correction is provided for one result of a paper by Shafaat [1], and it is proven that the set of Schreier varieties of algebras 〈A;⁺1〉 with two unary operations has the cardinality of the continuum.     

Congruence-distributive varieties of algebras

A survey of results on congruence-distributive varieties of algebras, concentrated on two directions: 1) the construction of algebras of congruence-distributive varieties and the structure of the variety itself, 2) questions connected with formalized (equational, elementary, etc.) theories of such varieties. In addition the survey includes papers connected with congruence-distributive varieties of algebras reviewed in RZh Matematika during the period 1976–1985.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 26, pp. 45–83, 1988.     

Representable varieties of algebras

Translated from Algebra i Logika, Vol. 28, No. 2, pp. 127–143, March–April, 1989.     

Universal varieties of quasi-Stone algebras

In this paper, we are mainly concerned with semicontinuity of complete lattices and their distributive reflections, introduced by Rav in 1989. We prove that for a complete lattice L, the distributive reflection L_d is isomorphic to the lattice of all radicals determined by principal ideals of L in the set-inclusion order, obtaining a method to depict the distributive reflection of a given lattice. It is also proved that if a complete lattice L is semicontinuous and every semiprime element x ∈ L is the largest in d(x), then L_d is continuous whenever the distributive reflector d is Scott continuous. We construct counterexamples to confirm a conjecture and solve two open problems posed by Zhao in 1997.     

Rigidity of free algebras in varieties of lie algebras

In this paper the main result is the rigidity of varietally free Lie algebras within the varieties where they are free. In fact, these algebras are usually free in some larger varieties, such as various types of the commutator varieties, and this is demonstrated in this paper as well. A related paper is [2] where, among some other results, we claimed the rigidity of free metabelian algebras within the respective variety of Lie algebras.