Non-modular varieties of semimodular lattices with a spanning \(M_ 3\) (Q762510)

There are several lattice constructions which go back to the tensor product \(A\otimes B\) in the category of join semilattices with zero (the morphisms are zero-preserving join homomorphisms). The author considers the tensor product of modular lattices to obtain special semimodular lattices. L is a lattice with spanning \(M_ 3\) if L contains 5 nullary operations a,b,c,0,1 so that \(a\wedge b=a\wedge c=b\wedge c=0\) and \(a\vee b=a\vee c=b\vee c=1\). The main results are: 1. For any bounded modular lattice B, \(M_ 3\otimes B\) is isomorphic to the lattice of all 1- preserving meet homomorphisms from \(M_ 3\) into B, and it is a semimodular lattice with a spanning \(M_ 3\). 2. Let \({\mathcal K}\) be a variety of modular lattices, \({\mathcal K}_ 3\) is the class of all lattices L with 5 nullary operations isomorphic to \(M_ 3\otimes B\) for some B in \({\mathcal K}\). Then \({\mathcal K}_ 3\) is a variety of semimodular lattices with spanning \(M_ 3\). 3. As category \({\mathcal K}_ 3\) is equivalent to \({\mathcal K}\), moreover \({\mathcal K}_ 3\) is finitely based iff \({\mathcal K}\) is finitely based. 4. If A is a bounded semimodular lattice and D is a bounded distributive lattice then \(A\otimes D\) is semimodular. 5. \(M_ 4\otimes M_ 4\) is not semimodular. 6. \(M_ 3\otimes M_ 3\) is not supersolvable (a lattice is supersolvable if it contains a maximal chain of modular elements). Some open problems are listed.