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Journal of Applied Mathematics and Computation

DOI：http://dx.doi.org/10.26855/jamc.2023.06.001

# The Substitution Sum of Boolean Matrices and the Substitution Product of Lattices

Zicheng Xie1, Congwen Luo1,2,*

1Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China.

2Department of Mathematics, College of Science, China Three Gorges University, Yichang, China.

*Corresponding author: Congwen Luo

Date: May 9,2023　Hits: 246

### Abstract

In this paper, we introduced the row and column space lattice of Boolean matrices. We prove that the row space lattice of the direct sum of Boolean matrices is isomorphic to the direct product of their row space lattice. We define the substitution of Boolean matrices and the substitution product with the lattice of the row and column space. And we prove that the rows and columns space lattice of the substitution sum of Boolean matrices is isomorphic to the substitution product of rows and columns space lattices of Boolean matrices. Then we gave some properties of the substitution product of lattice Spaces in row and column Spaces. Also, we establish the connection between Boolean matrices and finite lattices in a new way. With Boolean matrices, we can transform abstract algebraic operations into matrix operations. We believe that matrix representation will become an important tool in the study of lattice theory.

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### Full-Text HTML

The Substitution Sum of Boolean Matrices and the Substitution Product of Lattices

How to cite this paper: Zicheng Xie, Congwen Luo. (2023) The Substitution Sum of Boolean Matrices and the Substitution Product of Lattices. Journal of Applied Mathematics and Computation7(2), 202-210.

DOI: http://dx.doi.org/10.26855/jamc.2023.06.001