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...

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Bibliographic Details
Published inMathematics (Basel) Vol. 10; no. 7; p. 1143
Main Authors Guo, Hongfeng, Xing, Bing, Ming, Ziwei, Feng, Jun-E
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.04.2022
Subjects
Algebra
Algorithms
Boolean
Control theory
Intersections
Mathematics
Matrix algebra
Representations
semi-tensor product
structure matrices
Tensors
the number of topologies
Topology
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math10071143