w-independent quasi-equational basis of differential groupoids
DOI:
https://doi.org/10.26577/jmmcs-2017-3-465Keywords:
quasiidentity, quasivariety, quasi-equational bases, independent quasi-equational bases, differential groupoidsAbstract
The search for the solutions of the finite basis problem was and still is under the influence of
the problem of Alfred Tarski (Tarski, 1966: 275–288), who asked in 1966: if there is exists an
algorithm for deciding whether the quasi-equational theory of a finite set of finite algebras which
is assumed additionally to be equational, is finitely based. Tarski’s problem has been solved in
the negative by Ralph McKenzie (McKenzie, 1996: 49–104). The negative solution of Tarski’s
problem actually makes the finite basis problem more interesting and worthy of continued effort.
If Tarski’s problem had had a positive solution, the status of the finite basis problem would be
totally different. It would probably still exist, however, the main stream of the scientific effort would
go toward improving the known algorithms and classifying their complexity. However, the main
stream of the scientific effort would go toward improving the known algorithms and classifying
their complexity. We want to stress that the analogue of Tarski’s problem for the quasi-equational
theories that are not equational is unsolved. There is, however, a common believe that it will
also have a negative answer. It has been established at the end of the 1990-ties by a number of
researchers marvelous proof for the negative solution of Tarski’s problem cannot be easily modified
for quasi-equational theories. So today independent bases of quasiidentities have been found for
many classes of algebras and models. We will note that recently in work (Kravchenko, 2017с) has
found the general and sufficient condition for existence of a continuum of quasivarieties without
independent basis of quasiidentities, but with w - independent bases of quasiidentities. However
differential gruppoids aren’t sutisfied to these conditions and in this work we continued to study
the an independent quasiaquational basis of quasivarieties of differential gruppoids. The main
result is exists a continuum of quasivarieties of differential gruppoid which have w - independent
basis of quasiidentities.
References
III."Siberian Electronic Mathematical Reports. 14 (2017): 252–263.
[2] Belkin, Valerij. "Some lattices of quasivarieties of algebras."Algebra and logic. 5 (1976): 7–13.
[3] Budkin, Aleksandr. "Independent axiomatizability of quasivarieties of generalized solvable groups."Algebra and Logic. 25
(1986): 155–166.
[4] Budkin, Aleksandr. "Independent axiomatizability of quasivarieties of solube groups."Algebra and Logic. 30 (1991): 81–
100.
[5] Bestsennyi, Igorj. "Quasiidentities of finite unary algebras."Algebra and Logic. 28 (1989):327–340.
[6] Gorbunov, Viktor. "Covers in lattices of quasivarieties and independent axiomatizability."Algebra and Logic. 16 (1977):
340–369.
[7] Gorbunov, Viktor. Algebraic Theory of Quasivarieties. New York: Plenum, 1998.
[8] Kartashov, Vladimir. "Quasivarieties of unary algebras with a finite number of cycles."Algebra and Logic. 19 (1980):
106–120.
[9] Kartashova, Anna. "Antivarieties of unars."Algebra and Logic. 50 (2011): 357–364
[10] Kravchenko, Aleksandr. "Complexity of quasivariety lattices for varieties of unary algebras. II."Siberian Electronic
Mathematical Reports. 13 (2016): 388–394.
[11] Kravchenko, Aleksandr, and Anvar Nurakunov, and Marina Schwidefsky. "On quasi-equational bases for differential
groupoids and unary algebras."manuscript, 2017.
[12] Kravchenko, Aleksandr, and Anvar Nurakunov, and Marina Schwidefsky. "Complexity of quasivariety lattices. I. Covers
and independent axiomatizability."manuscript, 2017.
[13] Kravchenko, Aleksandr, and Aleksandr Yakovlev. "Quasivarieties of graphs and independent axiomatizability."manuscript,
2017.
[14] McKenzie Rudolf. "Tarski’s finite basis problem is undecidable."International Journal of Algebra and Computation. V.
—1996. vol.6, P. 49-104.
[15] Maltsev, Anatolij. "Universally axiomatizable subclasses of locally finite classes of models."Siberian Math. J. 8 (1967):
764–770.
[16] Maltsev, Anatolij. Algebraic Systems. Berlin, Heidelberg: Springer-Verlag, 1973.
[17] Nurakunov, Аnvar. Quasi-indentities of relatively distributive and relatively cocontinuous quasivarieties of algebras.
Darmstadt: Arbeitstangung Allgemeine Algebra, 1995.
[18] Romanowska, Anna. and Barbara Roszkowska. "On some groupoid models."Demonstr. Math. — 1987. vol. 20, No. 1-2 —
P.277–290.
[19] Romanowska, Anna. "On some representations of groupoid models satisfying the reduction law."Demonstr. Math. — 1988.
vol. 21, No. 4. — P.277–290.
[20] Romanowska, Anna. and Barbara Roszkowska. "Representation of n-cyclic groupoids."Algebra Universalis — 1989. vol. 26,
No. 1— P.7–15.
[21] Sapir, Mark. "On the quasivarieties generated by finite semigroups."Semigroup Forum. 20 (1980): 73–88.
[22] Sapir, Mark. "The lattice of quasivarieties of semigroups."Algebra Universalis. 21 (1985): 172–180.
[23] Sizyi, Sergei. "Quasivarieties of graphs."Siberian Math. J. 35 (1994): 783–794.
[24] Smirnov, Dmitrij. "Varieties and quasivarieties of algebras."Colloquia Math. Soc. J. 29 (1977): 745–751.
[25] Tarski, Аlfred. "Equational logic and equational theories of algebras."Contrib. Math. Logic. 8 (1966): 275–288.
[26] Tropin, Mihail. "Finite pseudo-Boolean and topological algebras not having an independent basis of
quasiindentities."Algebra and Logic. 27 (1988): 79–99.
[27] Budkin, A.I. (1982). Nezavisimaya aksiomatiziruemost kvazimnogoobrazij grupp [Independent axiomatizability of
quasivarieties of group]. Mat. zametki - math. notes, vol., 31, no.6, pp.817–826.
[28] Maltsev A.I. (1939) O vkluchenyi assosiativnykh sistem v gruppy [About including of associative systems in groups]. Mat.
sbornik - Math. collection, vol. 6, no. 2, pp.187–189.
[29] Maltsev A.I. (1968) O nekotorykh pogranichnikh voprosakh algebry i matematicheskoi logiki [About some boundary
questions of algebra and Mathematical logic]. Trudy kongressa matematikov - works of mathematic congress (Moskva,
1966), М.: Мir, pp.217–231.
[30] Medvedev N.Y. (1985) O kvazimnogoobraziykh l-grupp i grupp [about quasivarieties of l-group and group]. Sib. Mat. J.
- Sib. Math. J. vol. 26, pp. 111–117.
[31] Rumyansev A.K. (1975) Nezavisimyie bazisy dlya kvazitozhdestv svobodnoi kantorovoi algebry [Independent basis for
quasiindentities of free kantor algebra]. Mat.sbornic. - Math. collection, vol. 98, pp. 130–142.
[32] Smirnov, D.M. (1969). Reshetki mnogoobrazij i svobodnye algebry [Varieties of lattices and free algebras]. Sib. Mat. Zh.
- Sib. Math. J., vol. 10, pp. 1144–1160.
[33] Smirnov, D.M. (1972). Kantorovy algebry s odnim porozhdayushchim [Cantor algebras with one generating. I]. Algebra
and Logic, vol. 10, pp. 61–75.
[34] Tumanov V.I. (1984) Konechnye reshetki c nezavisimymi bazisami kvazitozhdestv [Finite lattice with independent quasiequational
basises] Mat. zametki - Math. notes, vol.36 : 811–815.
[35] Fedorov A.N. (1986) Kvazitozhdestva svobodnoi 2-nilpotentnoi gruppy [quasiidentities of free 2-nilpotent group] Mat.
zametki - Math. notes, vol. 40, pp. 590–597.
