Some problems in general algebra

Abstract

In the first part of the thesis we construct a finitely based variety, whose equational theory is undecidable, yet whose word problems are recursively solvable, which solves a problem stated by G. McNulty. The construction produces a discriminator variety with the aforementioned properties, starting from a class of structures in some multisorted language (which may include relations), axiomatized by a finite set of universal sentences in the given multisorted signature.

This result also makes present a common generalization of the earlier results obtained by B. Wells and A. Mekler, E. Belson, and S. Shelah.

In the second part of the dissertation the classification of finite graph M-algebras which have finite equational bases is given in terms of omitted induced subgraphs. The result is related to an earlier result obtained for finite graph algebras by K. Baker, G. McNulty, and H. Werner.