Journal of Applied Mathematics and Computation

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

Date: May 9,2023 Hits: 246

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.

[1] P. Luksch, R. Wille. (1988). Formal concept analysis of paired comparisons. Classification and related methods of data analysis, 167-176.

[2] P. Luksch, R. Wille. (1987). Substitution decomposition of concept lattices. Contributions to general algebra, 5, 213-220.

[3] J. Stephan. (1991). Substitution products of lattices. Contributions to general algebra, 7, 213-220.

[4] Š. Schwarz. (1970). On the semigroup of binary relations on a finite set. Czechoslovak Mathematical Journal, 20, 632-679.

[5] K. A. Zaretskii. (1963). The semigroup of binary relations. Sbornik, 61, 291-305.

[6] K. A. Zaretskii. (1962). Regular elements of the semigroup of binary relations. Uspeki Mat. Nauk, 17, 177-179.

[7] K. H. Kim. (1982). Boolean matrix theory and applications. Dekker, New York.

[8] R. Belohlavek, J. Konecny. (2012). Row and Column Spaces of Matrices over Residuated Lattices. Fundamenta Informaticae, 115, 279-295.

[9] V. Marenich. (2012). Lattices of Matrix Rows and Matrix Columns: Lattices of Invariant Column Eigenvectors. World Scientific, Singapore.

[10] S. K. Nimbhorkar, D. B. Banswal. (2019). Generalizations of supplemented lattices. AKCE INTERNATIONAL JOURNAL OF GRAPHS AND COMBINATORICS, 16, 8-17.

[11] I. Chajda, H. Langer. (2020). Residuation in lattice effect algebras. FUZZY SETS AND SYSTEMS, 397, 168-178.

[12] A. Aviles, G. Martinez-Cervantes, J. D. R. Abellan, A. R. Zoca. (2022). Lattice Embeddings in Free Banach Lattices Over Lattices. Mathematical Inequalities & Applications, 25, 495-509.

[13] B. A. Davey, H. A. Priestley. (2002). Introduction to Lattices and Order. Cambridge University Press, Cambridge.

[14] B. Ganter, R. Wille. (1999). Formal Concept Analysis. Springer-Verlag, Berlin.

[15] S. Rudeanu. (2001). Lattice Functions and Equations. Springer-Verlag, Berlin.

**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 Computation*, **7**(**2**), 202-210.

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

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