Simplicial complexes with lattice structures

TL;DR

This paper explores the topological lattice structures on the geometric realizations of order complexes of finite lattices, revealing new properties and answering longstanding questions about specific lattice examples.

Contribution

It introduces a natural topological lattice structure on the geometric realization of order complexes of finite lattices and analyzes their properties and variants.

Findings

The geometric realization of the order complex of a finite lattice forms a topological lattice.

The structure on $ riangle(M_3)$ is modular but not distributive, answering a question of Walter Taylor.

A new construction for 'stitching' lattices along a chain is described.

Abstract

If $L$ is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex $\Delta(L)$ (definition recalled). Lattice-theoretically, the resulting object is a subdirect product of copies of $L$. We note properties of this construction and of some variants thereof, and pose several questions. For $M_3$ the $5$-element nondistributive modular lattice, $\Delta(M_3)$ is modular, but its underlying topological space does not admit a structure of distributive lattice, answering a question of Walter Taylor. We also describe a construction of "stitching together" a family of lattices along a common chain, and note how $\Delta(M_3)$ can be obtained as a case of this construction.