B. Plotkin, G. Zhitomirski, “Some logical invariants of algebras and logical relations between algebras”, Algebra i Analiz, 19:5 (2007), 214–245; St. Petersburg Math. J., 19:5 (2008), 829

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Algebra i Analiz, 2007, Volume 19, Issue 5, Pages 214–245 (Mi aa142)

This article is cited in 18 scientific papers (total in 18 papers)

Research Papers

Some logical invariants of algebras and logical relations between algebras

B. Plotkin^a, G. Zhitomirski^b

^a Einstein Institute of Mathematics, Edmond J. Safra Campus, Hebrew University of Jerusalem, Jerusalem, Israel
^b Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel

Full-text PDF (302 kB) Citations (18)

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Abstract: Let $\Theta$ be an arbitrary variety of algebras and $H$ an algebra in $\Theta$. Along with algebraic geometry in $\Theta$ over the distinguished algebra $H$, a logical geometry in $\Theta$ over $H$ is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras $H\in\Theta$ are introduced and logical relations between different $H_1$ and $H_2$ in $\Theta$ 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.

Keywords: Variety of algebras, algebraic geometry, logical geometry.

Received: 15.05.2007

English version:
St. Petersburg Mathematical Journal, 2008, Volume 19, Issue 5, Pages 829–852
DOI: https://doi.org/10.1090/S1061-0022-08-01023-6

Bibliographic databases:

Document Type: Article

MSC: 03G25

Language: English

Citation: B. Plotkin, G. Zhitomirski, “Some logical invariants of algebras and logical relations between algebras”, Algebra i Analiz, 19:5 (2007), 214–245; St. Petersburg Math. J., 19:5 (2008), 829–852

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/aa142

https://www.mathnet.ru/eng/aa/v19/i5/p214

This publication is cited in the following 18 articles:

Kanel-Belov A., Chilikov A., Ivanov-Pogodaev I., Malev S., Plotkin E., Yu J.-T., Zhang W., “Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry”, Mathematics, 8:10 (2020), 1694
Aladova E., “Geometric View on Homogeneous Groups”, Groups, Algebras and Identities, Contemporary Mathematics, 726, ed. Plotkin E., Amer Mathematical Soc, 2019, 77–86
E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraic geometry over algebraic structures. VIII. Geometric equivalences and special classes of algebraic structures”, J. Math. Sci., 257:6 (2021), 797–813
A. G. Pinus, “Ob elementarnoi geometrii universalnykh algebr i ob ekvivalentnosti klonov otnositelno etoi geometrii”, Sib. elektron. matem. izv., 15 (2018), 332–337
Zhitomirski G., “Types of Points and Algebras”, Int. J. Algebr. Comput., 28:8, SI (2018), 1717–1730
E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraic geometry over algebraic structures. VI. Geometric equivalence”, Algebra and Logic, 56:4 (2017), 281–294
A. G. Pinus, “On the logical equivalence of functional clones”, Siberian Math. J., 58:4 (2017), 672–675
E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Universal geometrical equivalence of the algebraic structures of common signature”, Siberian Math. J., 58:5 (2017), 801–812
A. G. Pinus, “Ob odnom iz logicheskikh zamykanii na universalnykh algebrakh”, Sib. elektron. matem. izv., 12 (2015), 698–703
B. Plotkin, E. Plotkin, G. Zhitomirskii, “Type of a point in Universal Geometry and in Model Theory”, Algebra Discrete Math., 19:1 (2015), 87–100
Plotkin B., “Algebraic Logic and Logical Geometry in Arbitrary Varieties of Algebras”, Group Theory, Combinatorics, and Computing, Contemporary Mathematics, 611, eds. Morse R., NikolovaPopova D., Witherspoon S., Amer Mathematical Soc, 2014, 151–167
Plotkin B., Aladova E., Plotkin E., “Algebraic Logic and Logically-Geometric Types in Varieties of Algebras”, J. Algebra. Appl., 12:2 (2013), 1250146
A. G. Pinus, “The algebraic and logical geometries of universal algebras (a unified approach)”, J. Math. Sci., 185:3 (2012), 473–483
E. Aladova, A. A. Gvaramiya, B. I. Plotkin, “Logic in representations of groups”, Algebra and Logic, 51:1 (2012), 1–27
A. G. Pinus, “New algebraic invariants for definable subsets in universal algebra”, Algebra and Logic, 50:2 (2011), 146–160
B. I. Plotkin, “Isotyped Algebras”, Proc. Steklov Inst. Math., 278, suppl. 1 (2012), S91–S115
A. D. Maksimov, “Typical equivalence of linear groups and other algebraic systems”, J. Math. Sci., 183:3 (2012), 397–406
Plotkin B., “Some results and problems related to universal algebraic geometry”, Internat. J. Algebra Comput., 17:5-6 (2007), 1133–1164

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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