The geometric lattice of embedded subsets

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.