ON QUASI-IDENTITIES OF FINITE MODULAR LATTICES
Lutsak S.M. Voronina O.A. Nurakhmetova G.K.
26 September 2022al-Farabi Kazakh State National University
KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2022#115Issue 349 - 57 pp.
In 1970 R. McKenzie proved that any finite lattice has a finite basis of identities. However the similar result for quasi-identities is not true. That is, there is a finite lattice that has no finite basis of quasi-identities. The problem Which finite lattices have finite bases of quasi-identities? was suggested by V.A. Gorbunov and D.M. Smirnov. In 1984 V.I. Tumanov found a sufficient condition consisting of two parts under which a locally finite quasivariety of lattices has no finite (independent) basis of quasi-identities. Also he conjectured that a finite (modular) lattice has a finite basis of quasi-identities if and only if a quasivariety generated by this lattice is a variety. In general, the conjecture is not true. W. Dziobiak found a finite lattice that generates a finitely axiomatizable proper quasivariety. Tumanov’s problem is still unsolved for modular lattices. We construct a finite modular lattice that does not satisfy one of Tumanov’s conditions but the quasivariety generated by this lattice is not finitely based.
finite basis of quasi-identities , Lattice , quasivariety
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M. Kozybayev North Kazakhstan University, Petropavlovsk, Kazakhstan
M. Kozybayev North Kazakhstan University
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