Geodesic
Logically-geometrical similarity for algebras and models with the same identities
Trudy Instituta matematiki, Tome 23 (2015) no. 2, pp. 112-122.

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

The paper is related to the field which we call Universal Algebraic Geometry (UAG). All algebras under consideration belong to a variety of algebras $\Theta$. For an arbitrary $\Theta$ we construct a system of notions which lead to a bunch of new problems. As a rule, their solutions depend on the choice of specific $\Theta$. It can be the variety of groups $Grp$, the variety of associative or Lie algebras, etc. In particular, it can be the classical variety $Com-P$ of commutative and associative algebras with a unit over a field. For example, the paper concerns with the following general problem. For every algebra $H\in\Theta$ one can define the category of algebraic sets over $H$. Given $H_1$ and $H_2$ in $\Theta$, the question is what are the relations between these algebras that provide an isomorphism of the corresponding categories of algebraic sets. Similar problem stands with respect to the situation when algebras are replaced by models and categories of algebraic sets are replaced by categories of definable sets. The results on the stated problem are applicable to knowledge theory and, in particular, to knowledge bases. 
@article{TIMB_2015_23_2_a14,
     author = {E. Aladova and A. Gvaramia and B. Plotkin and E. Plotkin and T. Plotkin},
     title = {Logically-geometrical similarity for algebras and models with the same identities},
     journal = {Trudy Instituta matematiki},
     pages = {112--122},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2015_23_2_a14/}
}
TY  - JOUR
AU  - E. Aladova
AU  - A. Gvaramia
AU  - B. Plotkin
AU  - E. Plotkin
AU  - T. Plotkin
TI  - Logically-geometrical similarity for algebras and models with the same identities
JO  - Trudy Instituta matematiki
PY  - 2015
SP  - 112
EP  - 122
VL  - 23
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TIMB_2015_23_2_a14/
LA  - en
ID  - TIMB_2015_23_2_a14
ER  - 
%0 Journal Article
%A E. Aladova
%A A. Gvaramia
%A B. Plotkin
%A E. Plotkin
%A T. Plotkin
%T Logically-geometrical similarity for algebras and models with the same identities
%J Trudy Instituta matematiki
%D 2015
%P 112-122
%V 23
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TIMB_2015_23_2_a14/
%G en
%F TIMB_2015_23_2_a14
E. Aladova; A. Gvaramia; B. Plotkin; E. Plotkin; T. Plotkin. Logically-geometrical similarity for algebras and models with the same identities. Trudy Instituta matematiki, Tome 23 (2015) no. 2, pp. 112-122. http://geodesic.mathdoc.fr/item/TIMB_2015_23_2_a14/

[1] Mashevitzky G., Plotkin B., Plotkin E., “Automorphisms of categories of free algebras of varieties”, ERA of AMS, 8 (2002), 1–10 | MR

[2] Plotkin B., Plotkin E., “Multi-sorted logic and logical geometry: some problems”, Demonstratio Mathematica, XLVIII:4 (2015), 578–618 | MR

[3] Journal of Math. Sciences, 137:5 (2006), 5049– 5097 | DOI | MR | Zbl

[4] Hodges W., Model theory, Encyclopedia of Mathematics and its Applications, 42, Cambridge University Press, Cambridge, 1993 | MR | Zbl

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

[6] Plotkin B., Seven lectures on the universal algebraic geometry, 2002, 87 pp., arXiv: math/0204245 [math.GM]

[7] Plotkin B., Aladova E., Plotkin E., “Algebraic logic and logically-geometric types in varieties of algebras”, Journal of Algebra and its Applications, 12:2 (2013), 1250146, 23 pp. | DOI | MR | Zbl

[8] Halmos P. R., Algebraic logic, New York, 1969

[9] Aladova E., Plotkin E., Plotkin T., “Isotypeness of models and knowledge bases equivalence”, Mathematics in CS, 7:4 (2013), 421–438 | MR | Zbl

[10] Plotkin B., Universal algebra algebraic logic and databases, Kluwer Acad. Publ., 1994 | MR | Zbl

[11] Pinter C., “A simpler set of axioms for polyadic algebras”, Fundamenta Mathematicae, 79:3 (1973), 223–232 | MR | Zbl

[12] Pinter C., “A simple algebra of first-order logic”, Notre Dame journal of Formal Logic, XIV:14 (1973), 361–366 | DOI | MR | Zbl

[13] Galler B., “Cylindric and polyadic algebras”, Proceedings of the American Mathematical Society, 8 (1957), 176–183 | DOI | MR | Zbl

[14] Mac Lane S., Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, New York–Berlin, 1971 | MR | Zbl

[15] Plotkin B., “Algebras with the same algebraic geometry”, Proceedings of the Steklov Institute of Mathematics, 242, 2003, 176–207 | MR | Zbl

[16] Aladova E., Gvaramia A., Plotkin B., Plotkin T., “Multi-sorted logic, models and logical geometry”, Fundamental and Applied Mathematics, 19:3 (2014), 5–22