Some logical invariants of algebras and logical relations between algebras
HTML articles powered by AMS MathViewer
- by B. Plotkin and G. Zhitomirski
- St. Petersburg Math. J. 19 (2008), 829-852
- DOI: https://doi.org/10.1090/S1061-0022-08-01023-6
- Published electronically: June 27, 2008
- PDF | Request permission
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.References
- C. C. Chang and H. J. Keisler, Model theory, Studies in Logic and the Foundations of Mathematics, Vol. 73, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0409165
- Rami Grossberg, Classification theory for abstract elementary classes, Logic and algebra, Contemp. Math., vol. 302, Amer. Math. Soc., Providence, RI, 2002, pp. 165–204. MR 1928390, DOI 10.1090/conm/302/05080
- Paul R. Halmos, Algebraic logic, Chelsea Publishing Co., New York, 1962. MR 0131961
- Leon Henkin, J. Donald Monk, and Alfred Tarski, Cylindric algebras. Part II, Studies in Logic and the Foundations of Mathematics, vol. 115, North-Holland Publishing Co., Amsterdam, 1985. MR 781930
- Philip J. Higgins, Algebras with a scheme of operators, Math. Nachr. 27 (1963), 115–132. MR 163940, DOI 10.1002/mana.19630270108
- Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
- Gilbert Baumslag, Alexei Myasnikov, and Vladimir Remeslennikov, Algebraic geometry over groups. I. Algebraic sets and ideal theory, J. Algebra 219 (1999), no. 1, 16–79. MR 1707663, DOI 10.1006/jabr.1999.7881
- Alexei Myasnikov and Vladimir Remeslennikov, Algebraic geometry over groups. II. Logical foundations, J. Algebra 234 (2000), no. 1, 225–276. MR 1799485, DOI 10.1006/jabr.2000.8414
- B. Plotkin, Varieties of algebras and algebraic varieties. part B, Israel J. Math. 96 (1996), no. part B, 511–522. MR 1433704, DOI 10.1007/BF02937320
- B. Plotkin, Algebras with the same (algebraic) geometry, Tr. Mat. Inst. Steklova 242 (2003), no. Mat. Logika i Algebra, 176–207; English transl., Proc. Steklov Inst. Math. 3(242) (2003), 165–196. MR 2054494
- B. I. Plotkin, Some concepts of algebraic geometry in universal algebra, Algebra i Analiz 9 (1997), no. 4, 224–248 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 4, 859–879. MR 1604318
- —, Seven lectures on the universal algebraic geometry, Preprint (2002), Arxiv:math, GM/0204245, 87pp.
- B. Plotkin, Varieties of algebras and algebraic varieties. Categories of algebraic varieties, Siberian Adv. Math. 7 (1997), no. 2, 64–97. Siberian Advances in Mathematics. MR 1481222
- —, Algebraic geometry in first order logic, Itogi Nauki i Tekhniki Ser. Sovrem. Mat. i Prilozhen., vol. 22, VINITI, Moscow, 2004, pp. 16–62; English transl., J. Math. Sci. (N.Y.) 137 (2006), no. 5, 5049–5097; http://arxiv.org/abs/math.GM/0312485.
- Boris Plotkin, Some results and problems related to universal algebraic geometry, Internat. J. Algebra Comput. 17 (2007), no. 5-6, 1133–1164. MR 2355690, DOI 10.1142/S0218196707003986
- B. Plotkin and T. Plotkin, An algebraic approach to knowledge base models informational equivalence, Acta Appl. Math. 89 (2005), no. 1-3, 109–134 (2006). MR 2220200, DOI 10.1007/s10440-005-9008-z
Bibliographic 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
- Received by editor(s): May 15, 2007
- Published electronically: June 27, 2008
- © Copyright 2008 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 829-852
- MSC (2000): Primary 03G25
- DOI: https://doi.org/10.1090/S1061-0022-08-01023-6
- MathSciNet review: 2381947
Dedicated: Dedicated to the centenary of D. K. Faddeev’s birth