## Some Mal'cev conditions for varieties of algebras.

##### Abstract

This dissertation deals with the classification of varieties according to
their Mal'cev properties. In general the so called Mal'cev-type theorems
illustrate an interplay between first order properties of a given class of
algebras and the lattice properties of the congruence lattices of algebras of
the considered class.
CHAPTER 1. A survey of some notational conventions, relevant definitions
and auxiliary results is presented. Several examples of less frequently
used algebras are given together with the important properties of some of
them. The term algebra T(X) and useful results concerning 'term' operations
are established. A K-reflection is defined and a connection between
a K-reflection of an algebra and whether a class K satisfies an identity of
the algebra is established.
CHAPTER 2. The Mal'cev-type theorems are presented in complete
detail for varieties which are congruence permutable, congruence distributive,
arithmetical, congruence modular and congruence regular. Several
examples of varieties which exhibit these properties are presented together
with the necessary verifications.
CHAPTER 3. A general scheme of algorithmic character for some
Mal'cev conditions is presented. R. Wille (1970) and A. F. Pixley (1972)
provided algorithms for the classification of varieties which exhibit strong
Mal'cev properties. This chapter is largely devoted to a modification of
the Wille-Pixley schemes. It must be noted that this modification is quite
different from all such published schemes. The results are the same as in
Wille's scheme but slightly less general than in Pixley's. The text presented
here, however is much simpler. As an example, the scheme is used
to confirm Mal'cev's original theorem on congruence permutable varieties.
Finally, the so-called Chinese var£ety is defined and Mal'cev conditions are
established for such a variety of algebras .
CHAPTER 4. A comprehensive survey of literature concerning Mal'cev
conditions is given in this chapter.