The lattice of embedded subsets

The lattice of embedded subsets

In cooperative game theory, games in partition function form are real-valued function on the set of so-called embedded coalitions, that is, pairs (S, π) where S is a subset (coalition) of the set N of players, and π is a partition of N containing S. Despite the fact that many studies have been devoted to such games, surprisingly nobody clearly defined a structure (i.e., an order) on embedded co...

The geometric lattice of embedded subsets

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...

Diffraction of weighted lattice subsets

A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice Γ inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniform lattice Dirac comb, and its diffraction measure is periodic, with the dual lattice Γ ∗ as lattice of periods. This state...

Symmetric Subsets of Lattice Paths

Let [n] = {0, 1, . . . , n}. A subset S of [n] is symmetric if S = g− S for a natural number g. We show that, for every f : [n] → [2n] with the restriction 1 ≤ f(i + 1) − f(i) ≤ 2 for all i < n, there is some S ⊂ [n] such that |S| ≥ 2 lnn − O(1), with the property that both S and f(S) are symmetric. We prove this result by finding a lower bound for the length of a symmetric pattern whose abelia...

Diffraction of Weighted Lattice Subsets