Automorphisms of categories of free algebras of varieties
Authors:
G. Mashevitzky, B. Plotkin and E. Plotkin
Journal:
Electron. Res. Announc. Amer. Math. Soc. 8 (2002), 1-10
MSC (2000):
Primary 08A35, 08CO5, 14A22, 14A99
DOI:
https://doi.org/10.1090/S1079-6762-02-00099-9
Published electronically:
March 13, 2002
MathSciNet review:
1911740
Full-text PDF Free Access
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Abstract: Let $\Theta$ be an arbitrary variety of algebras and let $\Theta ^0$ be the category of all free finitely generated algebras from $\Theta$. We study automorphisms of such categories for special $\Theta$. 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 $\Theta$.
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A Automorphisms of classical groups, Moscow, Mir, 1976.
BMR1 G. Baumslag, A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups, J. Algebra 219 (1999), 16-79.
BMR2 G. Baumslag, A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups, in “Algorithmic problems in groups and semigroups”, Birkhäuser, Boston, 1999, pp. 35–51.
Be1 A. Berzins, Geometric equivalence of algebras, International J. of Algebra and Computations 11:4 (2001), 447–456.
Be2 A. Berzins, Variety Grp-F is semiperfect, Preprint, Riga, 1998.
BPP A. Berzins, B. Plotkin, E. Plotkin, Algebraic geometry in varieties of algebras with the given algebra of constants, Journal of Math. Sciences 102:3 (2000), 4039–4070.
Cz A. Czerniakiewicz, Automorphisms of a free associative algebra of rank $2$. II, Trans. Amer. Math. Soc. 171 (1972), 309–315.
Co P. M. Cohn, Free rings and their relations, Academic Press, 1985.
DF J. Dyer, E. Formanek, The automorphism group of a free group is complete, J. London Math. Soc. (2) 11:2 (1975), 181–190.
For E. Formanek, A question of B. Plotkin about the semigroup of endomorphisms of a free group, Proc. Amer. Math. Soc. 130 (2002), 935–937.
GoSh R. Gobel, S. Shelah. Radicals and Plotkin’s problem concerning geometrically equivalent groups, Proc. Amer. Math. Soc. 130 (2002), 673–674.
HR L. K. Hua and I. Reiner, Automorphisms of the unimodular group, Trans. Amer. Math. Soc. 71 (1951), 331–348.
LS R. C. Lyndon, P. E. Schupp, Combinatorial group theory, Springer Verlag, 1977. ;
ML L. Makar-Limanov, The automorphisms of the free algebra with two generators, Funktsional. Anal. i Pril. 4:3 (1970), 107–108. (Russian)
Ma G. Mashevitzky, A question of B. Plotkin about automorphisms of the semigroup of endomorphisms of a free semigroup, Preprint, Ben Gurion Univ., 2001.
MR2 A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups II. Logical foundations, J. of Algebra 234 (2000), 225–276.
Na M. Nagata, On the automorphism group of $k[x,y]$, Lectures in Math., Kyoto Univ., Kinokuniya, Tokyo, 1972
Pl7 B. Plotkin, Seven lectures in universal algebraic geometry, Preprint, Hebrew University, Jerusalem, 2000.
Pl3 B. Plotkin, Algebraic logic, varieties of algebras and algebraic varieties, in Proc. Int. Alg. Conf., St. Petersburg, 1995, St. Petersburg, 1999, pp. 189–271.
PIsr B. Plotkin, Varieties of algebras and algebraic varieties, Israel. Math. Journal 96:2 (1996), 511–522.
PSib B. Plotkin, Varieties of algebras and algebraic varieties. Categories of algebraic varieties, Siberian Advances in Mathematics, Allerton Press, 7:2 (1997), 64–97.
PAA B. Plotkin, Some notions of algebraic geometry in universal algebra, Algebra i Analiz 9:4 (1997), 224–248; English transl., St. Petersburg Math. J. 9:4 (1998), 859–879.
ST B. Schein and B. Teclezghi, Endomorphisms of finite full transformation semigroups, Proc. Amer. Math. Soc. 126 (1998), 2579–2587.
SH J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen, Fundamenta Mathematica 28 (1936), 261–264.
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Additional Information
G. Mashevitzky
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, 84105, Israel
Email:
gmash@cs.bgu.ac.il
B. Plotkin
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, 91904, Israel
Email:
borisov@math.huji.ac.il
E. Plotkin
Affiliation:
Department of Mathematics, Bar Ilan University, Ramat Gan, 52900, Israel
Email:
plotkin@macs.biu.ac.il
Keywords:
Algebraic variety,
variety of algebras,
category,
free algebra,
automorphisms
Received by editor(s):
July 4, 2001
Published electronically:
March 13, 2002
Additional Notes:
This work was supported in part by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities — Center of Excellence Program and by Intas grant “Algebraic $K$-theory, groups and algebraic homotopy theory”.
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 2002
American Mathematical Society
