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Some logical invariants of algebras and logical relations between algebras


Authors: B. Plotkin and G. Zhitomirski
Original publication: Algebra i Analiz, tom 19 (2007), nomer 5.
Journal: St. Petersburg Math. J. 19 (2008), 829-852
MSC (2000): Primary 03G25
DOI: https://doi.org/10.1090/S1061-0022-08-01023-6
Published electronically: June 27, 2008
MathSciNet review: 2381947
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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.


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Additional Information

B. Plotkin
Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, Hebrew University of Jerusalem, 91904, Jerusalem, Israel
Email: plotkin@macs.biu.ac.il, borisov@math.huji.ac.il

G. Zhitomirski
Affiliation: Department of Mathematics, Bar-Ilan University, 52900, Ramat Gan, Israel

DOI: https://doi.org/10.1090/S1061-0022-08-01023-6
Keywords: Variety of algebras, algebraic geometry, logical geometry
Received by editor(s): May 15, 2007
Published electronically: June 27, 2008
Dedicated: Dedicated to the centenary of D. K. Faddeev’s birth
Article copyright: © Copyright 2008 American Mathematical Society

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