Universal algebraic geometry: syntax and semantics
Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 2, pp. 75-88.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we give a general insight into the ideas that make ground for the developing of universal algebraic geometry and logical geometry. We specify the role of algebraic logic as one of the major instruments of the whole theory. The problem of the sameness of geometries of algebraic and definable sets for different algebras is considered as ans illuminating example how algebra, geometry, model theory, and algebraic logic work together.
@article{FPM_2020_23_2_a4,
     author = {A. Gvaramia and B. Plotkin and E. Plotkin},
     title = {Universal algebraic geometry: syntax and semantics},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {75--88},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2020_23_2_a4/}
}
TY  - JOUR
AU  - A. Gvaramia
AU  - B. Plotkin
AU  - E. Plotkin
TI  - Universal algebraic geometry: syntax and semantics
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2020
SP  - 75
EP  - 88
VL  - 23
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2020_23_2_a4/
LA  - ru
ID  - FPM_2020_23_2_a4
ER  - 
%0 Journal Article
%A A. Gvaramia
%A B. Plotkin
%A E. Plotkin
%T Universal algebraic geometry: syntax and semantics
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2020
%P 75-88
%V 23
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2020_23_2_a4/
%G ru
%F FPM_2020_23_2_a4
A. Gvaramia; B. Plotkin; E. Plotkin. Universal algebraic geometry: syntax and semantics. Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 2, pp. 75-88. http://geodesic.mathdoc.fr/item/FPM_2020_23_2_a4/

[1] Daniyarova E., Myasnikov A., Remeslennikov V., Algebraicheskaya geometriya nad algebraicheskimi sistemami, Izd-vo SO RAN, Novosibirsk, 2016

[2] Aladova E., “Syntax versus semantics in knowledge bases. I”, Int. J. Algebra Comput., 28:8 (2018), 1385–1404 | DOI | MR

[3] Chang C. C., Keisler H. J., Model Theory, North-Holland, 1973 | MR | Zbl

[4] Gvaramia A., “Halmos algebras and axiomatizable classes of quasigroups”, Usp. Mat. Nauk, 40:4 (1985), 215–216 | MR

[5] Halmos P. R., Algebraic Logic, New York, 1969 | MR

[6] Mac Lane S., Categories for the Working Mathematician, Grad. Texts Math., 5, Springer, Berlin, 1971 | MR | Zbl

[7] Malcev A. I., Algebraic Systems, Springer, Berlin, 1973 | MR

[8] Marker D., Model Theory: An Introduction, Springer, Berlin, 2002 | MR | Zbl

[9] Plotkin B., Universal Algebra, Algebraic Logic and Databases, Kluwer Academic, 1994 | MR | Zbl

[10] Plotkin B., Seven Lectures on the Universal Algebraic Geometry, 2002, arXiv: math/0204245 [math.GM] | MR

[11] Plotkin B., “Algebras with the same algebraic geometry”, Proc. Steklov Inst. Math., 242 (2003), 176–207 | MR | Zbl

[12] Plotkin B., “Algebraic geometry in first order logic”, J. Math. Sci., 137:5 (2006), 5049–5097 | DOI | MR | Zbl

[13] Plotkin B. I., “Isotyped algebras”, Proc. Steklov Inst. Math., 287 (2012), 91–115 | DOI

[14] Plotkin B., Aladova E., Plotkin E., “Algebraic logic and logically-geometric types in varieties of algebras”, J. Algebra Its Appl., 12:2 (2013), 1250146 | DOI | MR | Zbl

[15] Plotkin B., Plotkin E., “Multi-sorted logic and logical geometry: some problems”, Demonstratio Math., 48:4 (2015), 577–618 | MR

[16] Zhitomirski G., “On types of points and algebras”, Int. J. Algebra Comput., 28:8 (2018), 1717–1730 | DOI | MR | Zbl