Algebraic Representation of Topologies on a Finite Set

Hongfeng Guo, Bing Xing, Ziwei Ming and Jun-E Feng
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Hongfeng Guo: School of Statistics and Mathematics, Shandong University of Finance and Economics, Jinan 250014, China
Bing Xing: School of Statistics and Mathematics, Shandong University of Finance and Economics, Jinan 250014, China
Ziwei Ming: School of Statistics and Mathematics, Shandong University of Finance and Economics, Jinan 250014, China
Jun-E Feng: School of Mathematics, Shandong University, Jinan 250100, China

Mathematics, 2022, vol. 10, issue 7, 1-11

Abstract: 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.

Keywords: the number of topologies; semi-tensor product; structure matrices (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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