In the literature two notions of the word problem for a variety occur. A variety has a decidable word problem if every finitely presented algebra in the variety has a decidable word problem. It has a uniformly decidable word problem if there is an algorithm which given a finite presentation produces an algorithm for solving the word problem of the algebra so presented. A variety is given with f...

We prove that in a locally finite variety that has definable principal congruences (DPC), solvable congruences are nilpotent, and strongly solvable congruences are strongly abelian. As a corollary of the arguments we obtain that in a congruence modular variety with DPC, every solvable algebra can be decomposed as a direct product of nilpotent algebras of prime power size.

Let Γ be a finitely generated linear group over a field of characteristic 0. Suppose that every solvable subgroup of Γ is polycyclic. Then any solvable subgroup of Γ is separable. This conclusion is false without the hypothesis that every solvable subgroup of Γ is polycyclic.

A problem is defined to be exactly solvable if its solution can be obtained by solving a sequence of polynomials using radicals. Therefore, if a problem is not exactly solvable, then we have to use approximation schemes for solving the problem. It has been proved that the shortest network problem in space is not exactly solvable even if the network spans only four points and even if the topolog...

‎we describe a new type of presentation that‎, ‎when consistent‎, ‎describes a polycyclic group‎. ‎this presentation is obtained by‎ ‎refining a series of normal subgroups with‎ ‎abelian sections‎. ‎these presentations can be described effectively in‎ ‎computer-algebra-systems like {scshape gap} or ‎{scshape magma}‎. ‎we study these ‎presentations and‎, ‎in particular‎, ‎we obtain consistency c...