01.01.06 (Mathematical logic, algebra, and number theory)
algebraic systems; lattices connected with algebraic systems; varieties and quasivarieties of algebraic systems.
The main results relate to the theory of quasivarieties of unars (i.e. algebras with one unary operation). It was proved that any finitely generated unar has an independent basis of quasi-identities. Besides, this basis is finite if the unar is finite. An algorithm to construct an independent basis of quasi-identities of any finitely generated unar was proposed. It was shown that there exists the continuum of quasivarieties of unars having no an independent basis of quasi-identities. Let $L(1)$ be the lattice of all quasivarieties of unars. We described all atoms of this lattice. It was shown that a free lattice on $\aleph_0$ generators can be embedded into the lattice $L(1)$. We gave necessary and sufficient conditions for a lattice $L(M)$ of all subquasivarieties of a quasivariety $M$ of unars to be distributive, semidistributive, Boolean, complemented, pseudocomplemented or Stone. It was proved that the set of all quasivarieties of unars is a semigroop with respect to Mal'cev multiplication. It was described the center of this semigroop and constructed a basis of identities holding in this semigroop and containing one variable.
Graduated from Faculty of Physics and Mathematics of Volgograd Pedagogical State Institute in 1960. Ph. D. thesis was defended in 1980. A list of my works contains about 40 titles.
Kartashov V. K. Kvazimnogoobraziya unarov // Matem. zametki, 1980, 27(1), 7–20.
Kartashov V. K. Kvazimnogoobraziya unarov s konechnym chislom tsiklov // Algebra i logika, 1980, 19(2), 173–193.
Kartashov V. K. O reshetkakh kvazimnogoobrazii unarov // Sib. matem. zhurn., 1985, 26(3), 49–62.
Kartashov V. K. Stroenie distributivnykh reshetok kvazimnogoobrazii unarov // Sb. trudov mezhdunar. algebraich. seminara, posvyasch. pamyati L .A. Skornyakova "Universalnaya algebra i ee prilozheniya", 2000, 90–99.
Kartashov V. K. O guppoidakh kvazimnogoobrazii unarnykh algebr // Sb. trudov mezhdunar. algebraich. seminara, posvyasch. pamyati L. A. Skornyakova "Universalnaya algebra i ee prilozheniya", 2000, 100–113.
V. K. Kartashov, A. V. Kartashova, V. N. Ponomarjov, “On conditions for distributivity or modularity of congruence lattices of commutative unary algebras”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 13:4(2) (2013), 52–57
V. K. Kartashov, “On some results and unsolved problems of the theory of unary algebras”, Chebyshevskii Sb., 12:2 (2011), 18–26
V. K. Kartashov, “Independent systems of generators and the Hopf property for unary algebras”, Diskr. Mat., 20:4 (2008), 79–84; Discrete Math. Appl., 18:6 (2008), 625–630
V. K. Kartashov, “On the finite axiomatizability of varieties of commutative unary algebras”, Fundam. Prikl. Mat., 14:6 (2008), 85–89; J. Math. Sci., 164:1 (2010), 56–59
V. K. Kartashov, “Quasivarieties of unars”, Mat. Zametki, 27:1 (1980), 7–20; Math. Notes, 27:1 (1980), 5–12