Wu Wan*, Congwen Luo
College of Science, China Three Gorges University, Yichang, Hubei, China.
*Corresponding author: Wu Wan
Firstly, this paper defines a Galois connection between the row and column spaces of Boolean matrices, establishes the basic theorem of the row-column space lattice of Boolean matrices, and proves that any finite lattice is isomorphic to the row-column space lattice of Boolean matrices. Secondly, according to the basic theorem of the row-column space lattice of Boolean matrices, some special Boolean matrices are considered, such as reflexive matrices, symmetric matrices, equivalent matrices, etc. The row-column space lattices of these special Boolean matrices are characterized. It is proved that the row-column space lattice of sym-metric matrix corresponds to Polarity lattice, the row-column space lattice of anti-symmetric matrix corresponds to finite orthogonal lattice, the row-column space lattice of row (column) permutation matrix corresponds to Boolean lattice, etc. Finally, the relationship between row (column) permutation matrix and equivalent matrix is studied.
 Kim K.H. (1982). Boolean matrix theory and applications. Marcel Dekker, New York.
 Belohlavek R. and Konecny J. (2012). Row and column spaces of matrices over residuated lattices. Fundamenta Informaticae, 115, 279-295.
 Pattison P.E. and Breiger R.L. (2002). Lattices and dimensional representations: matrix decompositions and ordering structures. Social Networks, 24, 423-444.
 Davey B.A. and Priestley H.A. (2002). Introduction to lattices and order. Cambridge University Press, Cambridge.
 Rudeanu S. (2001). Lattice functions and equations. Springer-Verlin, Berlin.
 Ganter B. and Wille R. (1999). Formal concept analysis: mathematical foundations. Springer-Verlag, New York.
 Ma H. and Zhang K.L. (2015). Two new equivalent conditions of orthomodular lattices. Fuzzy Systems and Mathematics, 29, 27-30.
 Belohlavek R. and Trnecka M. (2018). A new algorithm for Boolean matrix factorization which admits overcovering. Discrete Applied Mathematics, 249, 36-52.
 LiangL.F., Zhu K.J., and Lu S.J. (2020). BEM: Mining coregulation patterns in transcriptomics via Boolean matrix factorization. Bioinformatics, 36, 4030-4037.
 Li X.L., Wang J., and Kwong S. (2022). Boolean matrix factorization based on collaborative neurodynamic optimization with Boltzmann machines. Neural Networks, 153, 142-151.
Row-column Space Lattices of Some Special Boolean Matrices
How to cite this paper: Wu Wan, Congwen Luo. (2023) Row-column Space Lattices of Some Special Boolean Matrices. Journal of Applied Mathematics and Computation, 7(1), 167-176.