Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



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
Published electronically: June 27, 2008
MathSciNet review: 2381947
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [CK] C. C. Chang and H. J. Keisler, Model theory, Stud. Logic Found. Math., vol. 73, North-Holland Publ. Co., Inc., New York, 1973. MR 0409165 (53:12927)
  • [G] R. Grossberg, Classification theory for abstract elementary classes, Logic and Algebra (Yi Zhang, ed.), Contemp. Math., vol. 302, Amer. Math. Soc., Providence, RI, 2002, pp. 165-204. MR 1928390 (2003h:03053)
  • [H] P. R. Halmos, Algebraic logic, Chelsea Publ. Co., New York, 1962. MR 0131961 (24:A1808)
  • [HMT] L. Henkin, J. D. Monk, and A. Tarski, Cylindric algebras. Part II, Stud. Logic Found. Math., vol. 115, North-Holland Publ. Co., Amsterdam, 1985. MR 0781930 (86m:03095b)
  • [Hi] P. J. Higgins, Algebras with a scheme of operators, Math. Nachr. 27 (1963), 115-132. MR 0163940 (29:1239)
  • [M] S. MacLane, Categories for the working mathematician, Grad. Texts in Math., vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798 (50:7275)
  • [MR1] G. Baumslag, A. Myasnikov, and V. Remeslennikov, Algebraic geometry over groups. I, J. Algebra 219 (1999), no. 1, 16-79. MR 1707663 (2000j:14003)
  • [MR2] A. Myasnikov and V. Remeslennikov, Algebraic geometry over groups. II. Logical foundations, J. Algebra 234 (2000), 225-276. MR 1799485 (2001i:14001)
  • [P1] B. Plotkin, Varieties of algebras and algebraic varieties, Israel J. Math. 96 (1996), part B, 511-522. MR 1433704 (98c:08011)
  • [P2] -, Algebras with the same (algebraic) geometry, Trudy Mat. Inst. Steklov. 242 (2003), 176-207; English transl., Proc. Steklov Inst. Math. 2003, no. 3 (242), 165-196. MR 2054494 (2005d:08013)
  • [P3] -, Some notions of algebraic geometry in universal algebra, Algebra i Analiz 9 (1997), no. 4, 224-248; English transl., St. Petersburg Math. J. 9 (1998), no. 4, 859-879. MR 1604318 (98j:08003)
  • [P4] -, Seven lectures on the universal algebraic geometry, Preprint (2002), Arxiv:math, GM/0204245, 87pp.
  • [P5] -, Varieties of algebras and algebraic varieties. Categories of algebraic varieties, Siberian Adv. Math. 7 (1997), no. 2, 64-97. MR 1481222 (99a:08004)
  • [P6] -, 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;
  • [P7] -, Some results and problems related to universal algebraic geometry, Internat. J. Algebra Comput. 17 (2007), nos. 5-6, 1133-1164. MR 2355690
  • [PP] 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); abs/math.GM/0312428. MR 2220200 (2006m:08011)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 03G25

Retrieve articles in all journals with MSC (2000): 03G25

Additional Information

B. Plotkin
Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, Hebrew University of Jerusalem, 91904, Jerusalem, Israel

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

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

American Mathematical Society