Some logical invariants of algebras and logical relations between algebras

Plotkin, B.; Zhitomirski, G.

doi:10.1090/S1061-0022-08-01023-6

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