Subsets Of Finite Set Research Articles

Algebraic Representation of Topologies on a Finite Set

Since the 1930s, topological counting on finite sets has been an interesting work so as to enumerate the number of corresponding order relations on the sets. Starting from the semi-tensor product (STP), we give the expression of the relationship between subsets of finite sets from the perspective of algebra. Firstly, using the STP of matrices, we present the algebraic representation of the subset and complement of finite sets and corresponding structure matrices. Then, we investigate respectively the relationship between the intersection and union and intersection and minus of structure matrices. Finally, we provide an algorithm to enumerate the numbers of topologies on a finite set based on the above theorems.

Lattice of Keychains

In this paper we consider the set of all n + 1-tuples of real numbers, not necessarily all distinct, in the decreasing order from the unit interval under the usual ordering of real numbers, always including 1. Such n+1-tuples inherently arise as the membership values of fuzzy subsets and are called keychains. An natural equivalence relation is introduced on this set and the equivalence classes of keychains are studied here. The number of such keychains is finite and the set of all keychains is a lattice under the coordinate-wise ordering. Thus keychains are subchains of a finite chain of real numbers in the unit interval. We study some of their properties and give some applications to counting fuzzy subsets of finite sets.

Combinatorial properties of some classes of matrices over GF(2)

We study several classes of matrices of GF(2) constructed from lists of subsets of finite sets In this paper. We show that all matrices in these classes are representations of connected equicardinal matrix over GF(2). In Matrix terms, these irreducible (defined below) matrices all have the property that every minimal dependent set of column has the same cardinality over GF(2). This fact is shown directly in this paper by elementary matrix considerations. In a subsequent paper, we shall show that these classes of matrices are in fact the classes of canonical forms for all representations of nontrivial binary connected equicardinal matroids.