The M 3 [ D ] construction and n -modularity

Abstract

In 1968, Schmidt introduced the M3[D] construction, an extension of the five-element modular nondistributive lattice M3 by a bounded distributive lattice D, defined as the lattice of all triples \(\langle x,y,z \rangle \in D^3\) satisfying \( x\wedge y=x\wedge z=y\wedge z\). The lattice M3[D] is a modular congruence-preserving extension of D.¶ In this paper, we investigate this construction for an arbitrary lattice L. For every n > 0, we exhibit an identity \( {\bf\mu}_n\) such that \( {\bf\mu}_1\) is modularity and \({\bf\mu}_{n+1}\) is properly weaker than \({\bf\mu}_n\). Let Mn denote the variety defined by \({\bf\mu}_n\), the variety of n-modular lattices. If L is n-modular, then M3[L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, Id M3[L] \(\cong \)M3[Id L]. ¶ We provide an example of a lattice L such that M3[L] is not a lattice. This example also provides a negative solution to a problem of Quackenbush: Is the tensor product \( A\otimes B\) of two lattices A and B with zero always a lattice. We complement this result by generalizing the M3[L] construction to an M4[L] construction. This yields, in particular, a bounded modular lattice L such that M4\( \otimes \)L is not a lattice, thus providing a negative solution to Quackenbush’s problem in the variety M of modular lattices.¶ Finally, we sharpen a result of Dilworth: Every finite distributive lattice can be represented as the congruence lattice of a finite 3-modular lattice. We do this by verifying that a construction of Gratzer, Lakser, and Schmidt yields a 3-modular lattice.