On quasi-identities of finite modular lattices

Abstract

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 sufficient condition consisting of two parts under which the 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 finitely axiomatizable proper quasivariety. Tumanov's problem is still unsolved for modular lattices. We construct a finite modular lattice that does not satisfy to one of the Tumanov's conditions but quasivariety generated by this lattice is not finitely based.