Algebraic Geometry over Groups

Baumslag, Gilbert; Myasnikov, Alexei; Remeslennikov, Vladimir N.

doi:10.1007/978-1-4612-1388-8_3

Cited by 38 publications

(38 citation statements)

References 18 publications

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“…Indeed, take H = H 1 that is not weakly AG-Noetherian. There is an ultrapower H 2 of H such that H 1 and H 2 are not AG-equivalent (see [MR1,P4]). Therefore, H 1 and H 2 are not LG-equivalent.…”

Section: Coordinate Algebras the Category Lc θ (H) Letmentioning

confidence: 99%

Some logical invariants of algebras and logical relations between algebras

Plotkin

Zhitomirski

2008

St. Petersburg Math. J.

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Abstract. Let Θ be an arbitrary variety of algebras and H an algebra in Θ. Along with algebraic geometry in Θ over the distinguished algebra H, a logical geometry in Θ over H is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras H ∈ Θ are introduced and logical relations between different H 1 and H 2 in Θ are analyzed. The paper contains a brief review of ideas of logical geometry ( §1), the necessary material from algebraic logic ( §2), and a deeper introduction to the subject ( §3). Also, a list of problems is given. 0.1. Introduction. The paper consists of three sections. A reader wishing to get a feeling of the subject and to understand the logic of the main ideas can confine himself to §1. A more advanced look at the topic of the paper is presented in § §2 and 3.In §1 we give a list of the main notions, formulate some results, and specify problems. Not all the notions used in §1 are well formalized and commonly known. In particular, we operate with algebraic logic, referring to §2 for precise definitions. However, §1 is self-contained from the viewpoint of ideas of universal algebraic geometry and logical geometry.Old and new notions from algebraic logic are collected in §2. Here we define the Halmos categories and multisorted Halmos algebras related to a variety Θ of algebras.§3 is a continuation of §1. Here we give necessary proofs and discuss problems. The main problem we are interested in is what are the algebras with the same geometrical logic.The theory described in the paper has deep ties with model theory, and some problems are of a model-theoretic nature.We emphasize once again that §1 gives a complete insight on the subject, while §2 and §3 describe and decode the material of §1. §1. Preliminaries. General view 1.1. Main idea. We fix an arbitrary variety Θ of algebras. Throughout the paper we consider algebras H in Θ. To each algebra H ∈ Θ one can attach an algebraic geometry (AG) in Θ over H and a logical geometry (LG) in Θ over H.In algebraic geometry we consider algebraic sets over H, while in logical geometry we consider logical (elementary) sets over H. These latter sets are related to the elementary logic, i.e., to the first order logic (FOL).Consideration of these sets gives grounds to geometries in an arbitrary variety of algebras. We distinguish algebraic and logical geometries in Θ. However, there is very 2000 Mathematics Subject Classification. Primary 03G25.

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Section: Coordinate Algebras the Category Lc θ (H) Letmentioning

confidence: 99%

Some logical invariants of algebras and logical relations between algebras

Plotkin

Zhitomirski

2008

St. Petersburg Math. J.

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“…Our primary interest in automorphisms of categories has grown from the universal algebraic geometry (see [18,22,21,20,2,3,16], etc). In order to make the exposition self-contained we recall the necessary information.…”

Section: Geometric Motivationmentioning

confidence: 99%

Automorphisms of categories of free algebras of varieties

Mashevitzky

Plotkin

2002

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. Let Θ be an arbitrary variety of algebras and let Θ 0 be the category of all free finitely generated algebras from Θ. We study automorphisms of such categories for special Θ. The cases of the varieties of all groups, all semigroups, all modules over a noetherian ring, all associative and commutative algebras over a field are completely investigated. The cases of associative and Lie algebras are also considered. This topic relates to algebraic geometry in arbitrary variety of algebras Θ.

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“…In the classical case, algebraic geometry has been shown to be very useful in dealing with polynomial equations over fields. An analog of algebraic geometry over groups has been developed by G. Baumslag, A. Myasnikov and V. Remeslennikov in [2]. It provides the necessary topological machinery as well as a method for transcribing geometric notions into the language of pure group theory.…”

Section: Theorem 2 the Elementary Theory Of A Free Group Is Decidablementioning

confidence: 99%

“…It provides the necessary topological machinery as well as a method for transcribing geometric notions into the language of pure group theory. Following [2] and [14] we can use standard algebraic geometry notions such as algebraic sets, the Zariski topology, Noetherian domains, irreducible varieties, radicals and coordinate groups to organize an approach to finding a solution of Tarski's problem. Some of these ideas go back to R. Bryant [4], V. Guba [11], B. Plotkin [28] and E. Rips.…”

Section: Theorem 2 the Elementary Theory Of A Free Group Is Decidablementioning

confidence: 99%

Tarski’s problem about the elementary theory of free groups has a positive solution

Kharlampovich

Myasnikov

1998

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We prove that the elementary theories of all nonabelian free groups coincide and that the elementary theory of a free group is decidable. These results answer two old questions that were raised by A. Tarski around 1945.The object of this announcement is to sketch proofs of the following two theorems. Theorem 1. The elementary theories of all nonabelian free groups coincide. Theorem 2. The elementary theory of a free group is decidable.These theorems answer two old questions that were raised by A. Tarski around 1945. We recall that the elementary theory T h(G) of a group G is the set of all first order sentences in the language of group theory which are true in G. Notice that in the language of group theory every sentence is equivalent to a sentence of the following type:

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Algebraic Geometry over Groups

Abstract: Classical commutative algebra provides the underpinnings of classical algebraic geometry. In this paper we will describe, without any proofs, a theory for groups that parallels this commutative algebra and that, in like fashion, is the basis of what we term algebraic geometry over groups.

Cited by 38 publications

References 18 publications

Some logical invariants of algebras and logical relations between algebras

Some logical invariants of algebras and logical relations between algebras

Automorphisms of categories of free algebras of varieties

Tarski’s problem about the elementary theory of free groups has a positive solution

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