The geometric lattice of embedded subsets

TL;DR

This paper introduces a new geometric lattice structure for embedded subsets using a closure operator, showing it is isomorphic to the partition lattice of a set with one additional element.

Contribution

It proposes a mathematically rigorous geometric lattice model for embedded subsets based on closure operators, improving upon previous ad-hoc definitions.

Findings

The lattice is geometric and atomic, satisfying the Steinitz exchange axiom.

The lattice of embedded subsets is isomorphic to the partition lattice of a set with one more element.

The approach clarifies the structure of embedded subsets within the product of subset and partition lattices.

Abstract

This work proposes an alternative approach to the so-called lattice of embedded subsets, which is included in the product of the subset and partition lattices of a finite set, and whose elements are pairs consisting of a subset and a partition where the former is a block of the latter. The lattice structure proposed in a recent contribution relies on ad-hoc definitions of both the join operator and the bottom element, while also including join-irreducible elements distinct from atoms. Conversely, here embedded subsets obtain through a closure operator defined over the product of the subset and partition lattices, where elements are generic pairs of a subset and a partition. Those such pairs that coincide with their closure are precisely embedded subsets, and since the Steinitz exchange axiom is also satisfied, what results is a geometric (hence atomic) lattice given by a simple matroid (or combinatorial geometry) included in the product of the subset and partition lattices (as the partition lattice itself is the polygon matroid defined on the edges of a complete graph). By focusing on its M\"obius function, this geometric lattice of embedded subsets of a n-set is shown to be isomorphic to the lattice of partitions of a n+1-set.