Varieties Of Universal Algebras Research Articles

Identities in congruence lattices of universal algebras

A variety of universal algebras is constructed which contains an algebra whose congruence lattice is nonmodular and which is such that the congruence lattice of each algebra in the variety satisfies a certain identity.

On the compatibility of algebraic equations with extensions

AbstractIn this paper we investigate the following problem proposed by Lausch and Nöbauer: Let be a variety of universal algebras, B an algebra of and A a subalgebra of B. If a system of algebraic equations over A is solvable in , is it then solvable over B? We show that the answer is affirmative in certain varieties but negative in the general case.

SCHREIER VARIETIES OF LINEAR Ω-ALGEBRAS

A variety of universal algebras is called a Schreier variety if every subalgebra of any free algebra in that variety is also free in that variety. This paper gives a description of the Schreier varieties of linear Ω-algebras over an associative commutative ring, defined by systems of homogeneous identities. As a corollary to these results one obtains a description of all Schreier varieties of linear Ω-algebras over an infinite field (in particular, over a field of characteristic zero). These algebras include, in particular, nonassociative algebras. Bibliography: 25 items.

FREE COMPACT ALGEBRAS

The class of all compact topological algebras in any variety of universal algebras is closed under the operations of taking direct products, closed sub-algebras and factor-algebras. This fact allows of a beginning to a theory of free compact algebras modelled on well-known studies of free topological groups and algebras by A. A. Markov, M. I. Graev and A. I. Mal'cev. Bibliography: 17 items.