On quasi-identities of finite modular lattices. II

A. O. Basheyeva, Svetlana Lutsak

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS, 2023

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DOI 10.31489/2023m2/45-52

Data confidence 70 medium

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Abstract

The existence of a finite identity basis for any finite lattice was established by R. McKenzie in 1970, but the analogous statement for quasi-identities is incorrect. So, there is a finite lattice that does not have a finite quasi-identity basis and, V.A. Gorbunov and D.M. Smirnov asked which finite lattices have finite quasiidentity bases. In 1984 V.I. Tumanov conjectured that a proper quasivariety generated by a finite modular lattice is not finitely based. He also found two conditions for quasivarieties which provide this conjecture. We construct a finite modular lattice that does not satisfy Tumanov’s conditions but quasivariety generated by this lattice is not finitely based.