FUNDAMENTALNAYA 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

Abstract

View as HTML     View as gif image

Let $$Q be a variety of algebras. In every variety $$Q and every algebra $H$ from $$Q one can consider algebraic geometry in $$Q over $H$. We also consider a special categorical invariant $K$_Q(H) of this geometry. The classical algebraic geometry deals with the variety $$Q = Com-P of all associative and commutative algebras over the ground field of constants $P$. An algebra $H$ in this setting is an extension of the ground field $P$. Geometry in groups is related to the varieties $Grp$ and $Grp-G$, where $G$ is a group of constants. The case $Grp-F$, where $F$ is a free group, is related to Tarski's problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras $H$₁ and $H$₂ have the same geometry? Or more specifically, what are the conditions on algebras from a given variety $$Q that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) $K$_Q(H₁) and $K$_Q(H₂) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let $$Q⁰ be the category of all algebras $W=W(X)$ free in $$Q, where $X$ is finite. Consider the groups of automorphisms $Aut($Q⁰) for different varieties $$Q and also the groups of autoequivalences of $$Q⁰. The problem is to describe these groups for different $$Q.

Location: http://mech.math.msu.su/~fpm/eng/k04/k043/k04309h.htm
Last modified: June 2, 2005