The M\₃[D] construction and n-modularity

In 1968, E. T. Schmidt introduced the M\_3[D] construction, an extension of the five-element nondistributive lattice M\_3 by a bounded distributive lattice D, defined as the lattice of all triples $(x, y, z) \in D^3$ satisfying $x \mm y = x \mm z = y \mm z$. The lattice M\_3[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 Un such that U1 is modularity and Un+1 is properly weaker than Un. Let Mn denote the variety defined by Un, the variety of n-modular lattices. If L is n-modular, then M\_3[L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, $IdM\_3[L] \cong M\_3[Id L]$. We provide an example of a lattice L such that M\_3[L] is not a lattice. This example also provides a negative solution to a problem of R. W. 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 M\_3[L] construction to an M\_4[L] construction. This yields, in particular, a bounded modular lattice L such that $M\_4 \otimes L$ 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 R. P. 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 G. Gr\"{a}tzer, H. Lakser, and E. T. Schmidt yields a 3-modular lattice.