Fast Möbius Inversion in Semimodular Lattices and ER-labelable Posets

We consider the problem of fast zeta and Möbius transforms in finite posets, particularly in lattices. It has previously been shown that for a certain family of lattices, zeta and Möbius transforms can be computed in $O(e)$ elementary arithmetic operations, where $e$ denotes the size of the covering relation. We show that this family is exactly that of geometric lattices. We also extend the algorithms so that they work in $e$ operations for all semimodular lattices, including chains and divisor lattices. Finally, for both transforms, we provide a more general algorithm that works in $e$ operations for all ER-labelable posets.

Geometric Lattice Structure of Covering and Its Application to Attribute Reduction through Matroids

The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially greedy algorithm design which plays an important role in the reduction problems. Therefore, it is meaningful to combine coverings with geometric lattices to solve the optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to the issue of attribute reduction. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its atoms are studied and used to describe the lattice. Second, considering that all the closed sets of a finite matroid form a geometric lattice, we propose a dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reduction. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice, to study the attribute reduction issues of information systems.

Lattices Generated by Orbits of Subspaces under Finite Singular Orthogonal Groups II

Let𝔽q(2ν+δ+l)be a(2ν+δ+l)-dimensional vector space over the finite field𝔽q. In this paper we assume that𝔽qis a finite field of odd characteristic, andO2ν+δ+l, Δ(𝔽q)the singular orthogonal groups of degree2ν+δ+lover𝔽q. Letℳbe any orbit of subspaces underO2ν+δ+l, Δ(𝔽q). Denote byℒthe set of subspaces which are intersections of subspaces inℳ, where we make the convention that the intersection of an empty set of subspaces of𝔽q(2ν+δ+l)is assumed to be𝔽q(2ν+δ+l). By orderingℒby ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these latticesℒare geometric lattices.