Automorphisms of categories of free algebras of varieties

Mashevitzky, G.; Plotkin, B.; Plotkin, Eugene

doi:10.1090/s1079-6762-02-00099-9

Cited by 32 publications

(37 citation statements)

References 17 publications

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“…However, it is important to know for which varieties Θ the equivalence Cl G 1 = Cl G 2 ⇐⇒ K Θ (G 1 ) ∼ = K Θ (G 2 ) holds. (The reader may consult [20], [21], [18], and [22] for many interesting results regarding this problem.) The fundamental notion of geometric similarity of algebras, generalizing the notion of geometric equivalence, proves to be crucial in all investigations concerning this problem.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules∗

Katsov

Lipyanski

Plotkin

2007

Communications in Algebra

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In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ 0 ) of automorphisms of the category Θ 0 of finitely generated free algebras of Θ is of great importance. In this paper, semi-inner automorphisms are defined for the categories of free (semi)modules and free Lie modules; then, under natural conditions on a (semi)ring, it is shown that all automorphisms of those categories are semi-inner. We thus prove that for a variety R M of semimodules over an IBN-semiring R (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of Aut( R M 0 ) are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in [22]; in particular, for Artinian (Noetherian, P I-) rings R, or a division semiring R, all automorphisms of Aut( R M 0 ) are semi-inner. April 30, 2005April 30, . 1991 Mathematics Subject Classification. Primary 16Y60, 16D90,16D99, 17B01; Secondary 08A35, 08C05. Date:Key words and phrases. free module, free semimodule over semiring, free modules over Lie algebras, semi-inner automorphism.

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Section: Introductionmentioning

confidence: 99%

Automorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules∗

Katsov

Lipyanski

Plotkin

2007

Communications in Algebra

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“…This role was underlined in Proposition 3 of Section 2.1. The meaning of Reduction Theorem (see References [17][18][19][20]) was explained just after this proposition. Reduction Theorem reduces investigation of automorphisms of the whole category Θ 0 of free in the variety Θ algebras to studying the group Aut(End(W(X))) associated with a single object W(X) in Θ 0 .…”

Section: Plotkin's Problem: Automorphisms Of Endomorphism Semigroups mentioning

confidence: 99%

“…It has been shown in Reference [17], (cf., Proposition 3) that geometrical similarity of algebras is determined by the structure of the group Aut(Theta 0 ), where Θ 0 is the category of free finitely generated algebras of Θ. The latter problem is treated by means of Reduction Theorem (see References [17][18][19][20]). This theorem reduces investigation of automorphisms of the whole category Θ 0 of free algebras in Θ to studying the group Aut(End(W(X))) associated with W(X) in Θ 0 .…”

Section: Introductionmentioning

confidence: 99%

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Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

et al. 2020

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This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given.

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“…(For a detailed explanation of these links, a list of references and related problems, see the excellent paper by Mashevitzky et al . [17].) Given the large number of published papers, preprints, lectures and, especially, open problems that have recently appeared on this subjectprompted by the links to universal algebraic geometry-we can anticipate that, for years to come, many new papers will be written describing the group of automorphisms of End(A) for various varieties A.…”

Section: Introductionmentioning

confidence: 99%