## Contents Preface (2002)

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@MISC{02contentspreface,

author = {},

title = {Contents Preface},

year = {2002}

}

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### Abstract

and Related Areas, sponsored by the Minerva Foundation (Germany)

### Citations

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524 |
Categories for the working mathematician
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Citation Context ...ge. Morphisms in the knowledge category give links between knowledge. In particular, one can speak of isomorphic knowledge. The categorical approach also allows us to use ideas of monada and comonada =-=[ML]-=-. It turns out that this leads to some general views on enrichment and computation of knowledge. Enrichment of a structure can be associated with a suitable monada over a category, while the correspon... |

438 | Combinatorial group theory - Lyndon, Schupp - 1977 |

138 | Infinite Abelian Groups - Fuchs - 1973 |

70 | Algebraic logic - Halmos - 1956 |

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53 | Cyclic splittings of finitely presented groups and the canonical JSJ decomposition - Rips, Sela - 1997 |

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48 | The Metamathematics of Algebraic Systems - Malcev - 1971 |

31 | Decidability of the universal and positive theories of a free group - Makanin - 1991 |

31 | Quantum Groups and Noncommutative Geometry, Publ. du C.R.M.; Univ. de Montreal - Manin - 1988 |

31 | Noncommutative schemes - Rosenberg |

31 |
Diophantine geometry over groups
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Citation Context ...n mentioned. However, the principal thing is to consider situations in the various special Θ and special H ∈ Θ. The algebraic geometry in groups is on rise now, see [BMR], [KhM] [MR1], [MR2], [BMRo], =-=[Se]-=- and others. It is quite reasonable to expect the similar breakthrough in Lie algebras and semigroups. For the case of associative algebras over a field or over on some other algebra of constants it i... |

27 | Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems - Guba - 1986 |

21 | The automorphism group of a free group is complete - Dyer, Formanek - 1975 |

20 | Algebras with the same algebraic geometry, in
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Citation Context ...2] and uses ideas of the dissertation of V.A. Gorbunov (1996) [Gor]. For groups there are also beautiful solutions in [GoSh] and [BlG]. For associative and Lie algebras this result is also valid (see =-=[Pl11]-=- for details. For associative algebras it uses the result from [Li]). On the other hand, recall that in classical situation every extension L of the field P is geometrically equivalent to every its ul... |

18 |
Varieties of algebras and algebraic varieties. Categories of algebraic varieties
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Citation Context ...OTKIN Theorem 1. An isomorphism F : KΘ1 (H1) → KΘ2 (H2) is a correct isomorphism if and only if Φ : CΘ1 (H1) → CΘ2 (H2) is a correct isomorphism. 2. Isomorphism, similarity, equivalence. In the paper =-=[Pl5]-=- the following theorems are proved Theorem 2. The categories KΘ(H1) and KΘ(H2) is correctly isomorphic if and only if the algebras H1 and H2 are geometrically similar. Let now H1 and H2 be abelian gro... |

17 |
Lectures in general algebra
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Citation Context ...0 is the zero element in the additive group H and on the left side we have n times for 0. Such group-based algebras are called also Ω-groups. They were introduced by P. Higgins [Hi] in 1956, see also =-=[Ku]-=-, [Pl7]. A group is an Ω-group with empty set Ω, in rings the set Ω consists of a single multiplication, in modules over a ring R all elements of R belong to the set Ω. Groups over rings, considered b... |

15 |
Universal algebra, algebraic logic and databases
- Plotkin
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(Show Context)
Citation Context ...e signature Ω. This signature allows us to consider a variety of G-algebras. Identities of a variety of G-algebras are presented by identities of Θ and by the defining relations of the algebra G (see =-=[Pl1]-=- for details). A free in Θ G algebra W = W(X) has the form G ∗ W0(X), where W0(X) is the free in the Θ algebra over X, ∗ is the free product in Θ and the embedding iG : G → W(X) = G ∗ W0(X) follows fr... |

14 | Algebraic logic, varieties of algebras and algebraic varieties, in - Plotkin - 1995 |

14 | Existentially closed groups - Higman, Scott - 1988 |

13 |
The verbal topology of a group
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Citation Context ...ngruences. If the variety Θ G is faithfully Noetherian, then each of its faithful algebras (H, h) is geometrically Noetherian. There are many important results on geometrically Noetherian groups (see =-=[Br]-=-, [Guba], [BMR1], [BMRo] and others. There are also many interesting problems, especially for the case of G in Θ G . 7. Geometric stability. This notion was already defined and will be used and discus... |

12 |
Algebraic geometry over groups
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Citation Context ...the variety Θ G is faithfully Noetherian, then each of its faithful algebras (H, h) is geometrically Noetherian. There are many important results on geometrically Noetherian groups (see [Br], [Guba], =-=[BMR1]-=-, [BMRo] and others. There are also many interesting problems, especially for the case of G in Θ G . 7. Geometric stability. This notion was already defined and will be used and discussed in the next ... |

12 | The cohomology theory of abstract algebraic varieties - Grothendieck - 1958 |

12 | Some questions of group theory related to geometry. In Algebra VII, Combinatorial group theory, Applications to geometry - Grigorchuk, Kurchanov - 1993 |

11 | A Myasnikov, Irreducible affine varieties over a free group. II. Systems in triangular quasi-quadratic form and description of residually free groups - Kharlampovich - 1998 |

11 | 1975], Continuously variable sets: algebraic geometry=geometric logic, in H. E.Rose - Lawvere - 1973 |

11 |
Some notions of algebraic geometry in universal algebra, Algebra i Analiz 9 (4
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Citation Context ...equivalent. Consider, in particular, the case when Θ1 and Θ2 are equivalent categories. Consider also the cases Θ1 = Θ G1 , Θ2 = Θ G2 for the different algebras G1 and G2 in the given Θ. Note that in =-=[Pl6]-=- and [Pl8] there is an invariant approach (without equations) to the category of algebraic varieties and it is quite natural to proceed here from the idea of equivalence of categories. There are some ... |

9 | Geometrical equivalence of algebras - Berzins |

9 | Lectures on Linear Groups - O’MEARA - 1974 |

9 | Adjunction of Elements to Groups.” The - Neumann - 1943 |

9 |
Tsurkov, Geometrical equivalence of groups
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Citation Context ...ors on classes of algebras, used in the characterization of prevarieties. For any class X the class LSC(X) is a locally closed prevariety over X which is contained in the quasivariety, generated by X =-=[PPT]-=-. For every algebra H ∈ Θ, finitely generated algebras in the prevariety SC(H) are the algebras presented as coordinate algebras of algebraic sets over H. It follows from [MR2] that the class LSC(X) i... |

8 |
Radicals and Plotkin’s problem concerning geometrically equivalent groups
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Citation Context ... is true also for G-groups. This very important expected result is obtained in [MR2] and uses ideas of the dissertation of V.A. Gorbunov (1996) [Gor]. For groups there are also beautiful solutions in =-=[GoSh]-=- and [BlG]. For associative and Lie algebras this result is also valid (see [Pl11] for details. For associative algebras it uses the result from [Li]). On the other hand, recall that in classical situ... |

6 |
Two theorems about equationally Noetherian groups
- Baumslag, Myasnikov, et al.
- 1997
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Citation Context ...ety Θ G is faithfully Noetherian, then each of its faithful algebras (H, h) is geometrically Noetherian. There are many important results on geometrically Noetherian groups (see [Br], [Guba], [BMR1], =-=[BMRo]-=- and others. There are also many interesting problems, especially for the case of G in Θ G . 7. Geometric stability. This notion was already defined and will be used and discussed in the next lecture.... |

6 | Some notes on universal algebraic geometry, in: Algebra - Nikolova, Plotkin - 1998 |

6 |
Groups of automorphisms of algebraic systems
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- 1972
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Citation Context ...he zero element in the additive group H and on the left side we have n times for 0. Such group-based algebras are called also Ω-groups. They were introduced by P. Higgins [Hi] in 1956, see also [Ku], =-=[Pl7]-=-. A group is an Ω-group with empty set Ω, in rings the set Ω consists of a single multiplication, in modules over a ring R all elements of R belong to the set Ω. Groups over rings, considered by R.Lyn... |

5 | Some remarks on quasi-varieties of algebraic structures - Malcev - 1966 |

4 | Algebraic geometry over groups, in book “Algorithmic problems in groups and semigroups - Baumslag, Remeslennikov - 1999 |

4 |
with parametric exponents
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Citation Context ...group is an Ω-group with empty set Ω, in rings the set Ω consists of a single multiplication, in modules over a ring R all elements of R belong to the set Ω. Groups over rings, considered by R.Lyndon =-=[L2]-=- are also Ω-groups. In arbitrary Ω-group G we have the usual commutator [a, b] = −a −b+a+b = −a + a b , and ω-commutators for ω with n(ω) > 0. By definition we have: [a1, · · · , an; b1, · · · , bn; ω... |

4 | A note on algebraically closed groups - Neumann - 1952 |

3 | Algebraic theory of quasivarieties, Doctoral Thesis - Gorbunov - 1996 |

3 | Topics in noncommutive geometry, Princeton Univ - Manin - 1991 |

3 | Positive formulae on free groups - Merzljakov - 1966 |

3 | of algebras and algebraic varieties Israel - Plotkin - 1996 |

3 | Algebraic geometry in First Order Logic, Sovremennaja Matematika and Applications 22 (2004), 16–62 - Plotkin |

3 |
Infinitary quasi-identities and infinitary quasivarieties, Proc
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Citation Context ...r, not an axiomatized class. In this sense, the relation of geometric equivalence of algebras is not an axiomatizable relation. This relation is axiomatizable in terms of generalized quasi-identities =-=[Pl10]-=-. 3. Geometric similarity of algebras. This notion generalizes the notion of geometric equivalence of algebras, and, like the notion of geometric equivalence, is associated with the problem of isomorp... |

3 | On systems of equations in a free groups, Combinatorial and geometric group theory - Razborov - 1993 |

3 | Geometrical aspect of databases and knowledge bases, vol. 46, Algebra Universals - Plotkin, Plotkin - 2001 |

2 |
On geometric equivalence of groups
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- 2002
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Citation Context ...so for G-groups. This very important expected result is obtained in [MR2] and uses ideas of the dissertation of V.A. Gorbunov (1996) [Gor]. For groups there are also beautiful solutions in [GoSh] and =-=[BlG]-=-. For associative and Lie algebras this result is also valid (see [Pl11] for details. For associative algebras it uses the result from [Li]). On the other hand, recall that in classical situation ever... |

2 | categories and databases - Plotkin, Algebra |