Varieties generated by lattices of breadth two
A lattice \(L\) is called dismantlable if every non-empty sublattice of \(L\) contains a doubly irreducible (i.e. join-irreducible and meet- irreducible) element. It is shown that a dismantlable lattice is a lattice of breadth \(\leq 2\). Some relations between the class \({\mathcal B}\) of all lattices of breadth \(\leq 2\) and the class \({\mathcal D}\) of all dismantlable lattices, as well as the lattice varieties \({\mathbf V} ({\mathcal B})\) and \({\mathbf V} ({\mathcal D})\) generated by \({\mathcal B}\) and \({\mathcal D}\), resp., are studied in the paper. It is noted that \({\mathbf V} ({\mathcal B})\) satisfies \((x\vee y)\wedge (x\vee z)= x\vee ((x\vee y)\wedge (x\vee z)\wedge (y\vee z))\). The main results are as follows. A lattice of breadth \(\leq n\) is \(n\)-distributive (Proposition 2). For finite join- semidistributive (meet-semidistributive) lattice concepts of dismantlability and breadth \(\leq 2\) coincide (Proposition 1). On the other hand, the interval \([{\mathbf V} ({\mathcal D}), {\mathbf V} ({\mathcal B})]\) of the lattice of all lattice varieties contains continuum many varieties (Theorem 1), and all varieties of this interval are not finitely based (Theorem 2).
scientific article; zbMATH DE number 540214
scientific article