Let an arbitrary variety of algebras and the category of all free finitely generated algebras in that variety be given. In universal algebraic geometry over an arbitrary variety of algebras, the group of automorphisms of the category of free finitely generated algebras plays an important role. This paper is first in a series where we will deal with the group mentioned. Here we describe properties of automorphisms of the category of all free finitely generated algebras and distinguish two important subgroups: namely, the subgroup of inner automorphisms and the subgroup of strongly stable automorphisms.
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References
B. Plotkin, “Algebras with the same (algebraic) geometry,” Trudy Steklov Mat. Inst., 242, 176-207 (2003).
B. Plotkin, “Seven lectures on universal algebraic geometry,” Contemp. Math., 726, 143-215 (2019).
B. Plotkin and E. Plotkin, “Multi-sorted logic and logical geometry: Some problems,” Demonstr. Math., 48, No. 4, 577-619 (2015).
E. Yu. Daniyarova, A. G. Myasnikov, and V. N. Remeslennikov, Algebraic Geometry over Algebraic Structures [in Russian], SO RAN, Novosibirsk (2016).
B. I. Plotkin, “Some notions of algebraic geometry in universal algebra,” Alg. Analysis, 9, No. 4, 224-248 (1997).
R. Göbel and S. Shelah, “Radicals and Plotkin’s problem concerning geometrically equivalent groups,” Proc. Am. Math. Soc., 130, No. 3, 673-674 (2002).
A. Myasnikov and V. N. Remeslennikov, “Algebraic geometry over groups II. Logical foundations,” J. Alg., 234, No. 1, 225-276 (2000).
Y. Katsov, R. Lipyanski, and B. Plotkin, “Automorphisms of categories of free modules, free semimodules, and free Lie modules,” Comm. Alg., 35, No. 3, 931-952 (2007).
G. Mashevitzky, B. Plotkin, and E. Plotkin, “Automorphisms of categories of free algebras of varieties,” Electron. Res. Announc. Am. Math. Soc., 8, 1-10 (2002).
G. Mashevitzky, B. Plotkin, and E. Plotkin, “Automorphisms of the category of free Lie algebras,” J. Alg., 282, No. 2, 490-512 (2004).
G. Mashevitzky, B. M. Schein, and G. I. Zhitomirski, “Automorphisms of the endomorphism semigroup of a free inverse semigroup,” Comm. Alg., 34, No. 10, 3569-3584 (2006).
I. Shestakov and A. Tsurkov, “Automorphic equivalence of the representations of Lie algebras,” Alg. Discr. Math., 15, No. 1, 96-126 (2013).
A. Tsurkov, “Automorphic equivalence of many-sorted algebras,” Appl. Categ. Struct., 24, No. 3, 209-240 (2016).
S. Mac Lane, Categories for the Working Mathematician, Grad. Texts Math., 5, Springer-Verlag, New York (1971).
F. Kasch, Modules and Rings, London Math. Soc. Monogr., 17, Academic Press, New York (1982).
P. M. Cohn, “Some remarks on the invariant basis property,” Topology, 5, 215-228 (1966).
B. Plotkin and G. Zhitomirski, “Automorphisms of categories of free algebras of some varieties,” J. Alg., 306, No. 2, 344-367 (2006).
A. Tsurkov, “Automorphic equivalence of algebras,” Int. J. Alg. Comput., 17, Nos. 5/6, 1263-1271 (2007).
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Translated from Algebra i Logika, Vol. 61, No. 1, pp. 3-22, January-February, 2022. Russian DOI: https://doi.org/10.33048/alglog.2022.61.101.
To 95th birthday of my teacher B. I. Plotkin
Supported by the ISF grants 1623/16, 1994/20, and by Gelbart Institute for Mathematical Sciences, Department of Mathematics, Bar-Ilan University, Israel.
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Aladova, E.V. Automorphisms of the Category of Free Finitely Generated Algebras. Algebra Logic 61, 1–15 (2022). https://doi.org/10.1007/s10469-022-09671-1
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DOI: https://doi.org/10.1007/s10469-022-09671-1