Characterization of Birkhoff’s Conditions by Means of Cover-preserving and Partially Cover-preserving Sublattices

In the paper we investigate Birkhoff’s conditions (Bi) and (Bi∗). We prove that a discrete lattice L satisfies the condition (Bi) (the condition (Bi∗)) if and only if L is a 4-cell lattice not containing a cover-preserving sublattice isomorphic to the lattice S∗ 7 (the lattice S7). As a corollary we obtain a well known result of J. Jakub́ık from [6]. Furthermore, lattices S7 and S ∗ 7 are considered as so-called partially coverpreserving sublattices of a given lattice L, S7 L and S∗ 7 L, in symbols. It is shown that an upper continuous lattice L satisfies (Bi∗) if and only if L is a 4-cell lattice such that S7 6 L. The final corollary is a generalization of Jakub́ık’s theorem for upper continuous and strongly atomic lattices.