The M3[D] construction and n-modularity

Abstract.

In 1968, Schmidt introduced the M ₃[D] construction, an extension of the five-element modular nondistributive lattice M ₃ 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 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 ${\bf\mu}_n$ such that ${\bf\mu}_1$ is modularity and ${\bf\mu}_{n+1}$ is properly weaker than ${\bf\mu}_n$. Let M _n denote the variety defined by ${\bf\mu}_n$, 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, Id M 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 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 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 Grätzer, Lakser, and Schmidt yields a 3-modular lattice.