Varieties of lattices with geometric descriptions

TL;DR

This paper extends the embedding results of modular lattices to n-distributive lattices, showing their varieties are generated by finite members, which leads to decidability of their word problem and explores spatial properties.

Contribution

It generalizes embedding theorems to n-distributive lattices, proving their varieties are finitely generated and decidable, and constructs examples with specific spatial properties.

Findings

Every n-distributive lattice embeds into a strongly spatial lattice.

The variety of n-distributive lattices is generated by finite members.

A lattice is constructed that cannot embed into any algebraic and spatial lattice.

Abstract

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann, Pickering, and Roddy proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite, join-semidistributive variety.