I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2004, VOLUME 10, NUMBER 3, PAGES 181-197
Problems in algebra inspired by universal algebraic geometry
B. I. Plotkin
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Let be a variety of
In every variety and every
from one can consider
algebraic geometry in over .
We also consider a special categorical invariant of this
The classical algebraic geometry deals with the variety
of all associative and commutative algebras over the ground field of
An algebra in this setting is an
extension of the ground field .
Geometry in groups is related to the varieties and , where
a group of constants.
The case ,
where is a free group, is
related to Tarski's problems devoted to logic of a free group.
The described general insight on algebraic geometry in different
varieties of algebras inspires some new problems in algebra and
The problems of such kind determine, to a great extent, the
content of universal algebraic geometry.
For example, a general and natural problem is: When do algebras
and have the same
geometry? Or more specifically, what are the conditions on algebras
from a given variety that provide
the coincidence of their algebraic geometries? We consider two variants of
are isomorphic; 2) these categories are equivalent.
This problem is closely connected with the following general algebraic
Let be the
category of all algebras free
in , where is finite.
Consider the groups of automorphisms for
different varieties and also the groups
of autoequivalences of .
The problem is to describe these groups for different .
Last modified: June 2, 2005