scholarly journals On group automorphisms in universal algebraic geometry

2019 ◽  
Vol 11 (2) ◽  
pp. 115-121
Author(s):  
Artem N. Shevlyakov
Keyword(s):  
Algebraic Geometry ◽  
Open Problem ◽  
Study Group ◽  
Universal Algebraic Geometry ◽  
Group Automorphisms

Abstract In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.

Dimension in universal algebraic geometry

2014 ◽  
Vol 90 (1) ◽  
pp. 450-452 ◽  
Author(s):  
E. Yu. Daniyarova ◽  
A. G. Myasnikov ◽  
V. N. Remeslennikov
Keyword(s):  
Algebraic Geometry ◽  
Universal Algebraic Geometry

ALGEBRAIC GEOMETRY FOR MV-ALGEBRAS

2014 ◽  
Vol 79 (4) ◽  
pp. 1061-1091 ◽  
Author(s):  
LAWRENCE P. BELLUCE ◽  
ANTONIO DI NOLA ◽  
GIACOMO LENZI
Keyword(s):  
Algebraic Geometry ◽  
Polynomial Functions ◽  
Algebraic Sets ◽  
Mv Algebra ◽  
Geometric Objects ◽  
Universal Algebraic Geometry ◽  
Mv Algebras

AbstractIn this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

Universal algebraic geometry

2011 ◽  
Vol 84 (1) ◽  
pp. 545-547 ◽  
Author(s):  
E. Yu. Daniyarova ◽  
A. G. Myasnikov ◽  
V. N. Remeslennikov
Keyword(s):  
Algebraic Geometry ◽  
Universal Algebraic Geometry

scholarly journals Compactness Conditions in Universal Algebraic Geometry

2016 ◽  
Vol 55 (2) ◽  
pp. 146-172 ◽  
Author(s):  
P. Modabberi ◽  
M. Shahryari
Keyword(s):  
Algebraic Geometry ◽  
Universal Algebraic Geometry

scholarly journals AUTOMORPHISMS OF THE CATEGORY OF THE FREE NILPOTENT GROUPS OF THE FIXED CLASS OF NILPOTENCY

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1273-1281 ◽  
Author(s):  
A. TSURKOV
Keyword(s):  
Algebraic Geometry ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Free Algebras ◽  
Finitely Generated ◽  
Identity Automorphism ◽  
Universal Algebraic Geometry ◽  
Inner Automorphisms

This paper is motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The key problem is to study how far the group Aut Θ0 of all automorphisms of the category Θ0 is from the group Inn Θ0 of inner automorphisms of Θ0 (see [7, 10] for details). Recall that an automorphism ϒ of a category 𝔎 is inner, if it is isomorphic as a functor to the identity automorphism of the category 𝔎. Let Θ = 𝔑d be the variety of all nilpotent groups whose nilpotency class is ≤ d. Using the method of verbal operations developed in [8, 9], we prove that every automorphism of the category [Formula: see text], d ≥ 2 is inner.

SOME RESULTS AND PROBLEMS RELATED TO UNIVERSAL ALGEBRAIC GEOMETRY

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1133-1164 ◽  
Author(s):  
BORIS PLOTKIN
Keyword(s):  
Algebraic Geometry ◽  
Point Of View ◽  
Variety Of Algebras ◽  
Geometric Properties ◽  
Algebraic Sets ◽  
The Subject ◽  
Universal Algebraic Geometry ◽  
Future Work ◽  
The Given

In universal algebraic geometry (UAG), some primary notions of classical algebraic geometry are applied to an arbitrary variety of algebras Θ and an arbitrary algebra H ∈ Θ. We consider an algebraic geometry in Θ over the distinguished algebra H and we also analyze H from the point of view of its geometric properties. This insight leads to a system of new notions and stimulates a number of new problems. They are new with respect to algebra, algebraic geometry and even with respect to the classical algebraic geometry. In our approach, there are two main aspects: the first one is a study of the algebra H and its geometric properties, while the second is focused on studying algebraic sets and algebraic varieties over a "good", particular algebra H. Considering the subject from the second standpoint, the main goal is to get forward as far as possible in a classification of algebraic sets over the given H. The first approach does not require such a classification which is itself an independent and extremely difficult task. We also consider some geometric relations between different H1 and H2 in Θ. The present paper should be viewed as a brief review of what has been done in universal algebraic geometry. We also give a list of unsolved problems for future work.

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