Lattices Associated with a Finite Vector Space

Author(s)
Mengtian Yue

ABSTRACT

Let be a n-dimensional row vector
space over a finite field For , let be a *d*- dimensional subspace of . denotes the set
of all the spaces which are the subspaces of and not the
subspaces of except . We define the partial order on by ordinary
inclusion (resp. reverse inclusion), and then is a poset,
denoted by (resp. ). In this paper we show that both and are finite
atomic lattices. Further, we discuss the geometricity of and , and obtain their characteristic polynomials.

Cite this paper

Yue, M. (2014) Lattices Associated with a Finite Vector Space.*Applied Mathematics*, **5**, 672-676. doi: 10.4236/am.2014.54064.

Yue, M. (2014) Lattices Associated with a Finite Vector Space.

References

[1] Aigner, M. (1979) Combinatorial Theory. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-1-4615-6666-3

[2] Wan, Z. and Huo, Y. (2004) Lattices Generated by Transitive Sets of Subspaces Under Finite Classical Groups. 2nd Edition, Science Press, Beijing.

[3] Liu, Y.H. (2011) Special Lattice of Rough Algebras. Applied Mathematics, 2, 1522-1524.

http://dx.doi.org/10.4236/am.2011.212215

[4] Huo, Y.J. and Wan, Z.-X. (2001) On the Geomericity of Lattices Generated by Orbits of Subspaces under Finite Classical Groups. Journal of Algebra, 243, 339-359.

http://dx.doi.org/10.1006/jabr.2001.8819

[5] Gao, Y. and You, H. (2003) Lattices Generated by Orbits of Subspaces under Finite Singular Classical Groups and Its Characteristic Polynomials. Communications in Algebra, 31, 2927-2950. http://dx.doi.org/10.1081/AGB-120021900

[6] Wang, K.S. and Guo, J. (2009) Lattices Generated by Two Orbits of Subspaces Under Finite Classical Groups. Finite Fields and Their Applications, 15, 236-245. http://dx.doi.org/10.1016/j.ffa.2008.12.008

[7] Wang, K.S. and Li, Z.T. (2008) Lattices Associated with Vector Spaces Over a Finite Field. Linear Algebra and Its Applications, 429, 439-446. http://dx.doi.org/10.1016/j.laa.2008.02.035

[8] Guo, J., Li, Z.T. and Wang, K.S. (2009) Lattices Associated with Totally Isotropic Subspaces in Classical Spaces. Linear Algebra and Its Applications, 431, 1088-1095.

http://dx.doi.org/10.1016/j.laa.2009.04.009

[9] Gao, S.G., Guo, J. and Liu, W. (2007) Lattices Generated by Strongly Closed Subgraphs in d-Bounded Distance-Regular Graphs. European Journal of Combinatorics, 28, 1800-1813.

http://dx.doi.org/10.1016/j.ejc.2006.05.011

[10] Guo, J. and Gao, S.G. (2008) Lattices Generated by Join of Strongly Closed Subgraphs in d-Bounded Distance-Regular Graphs. Discrete Mathematics, 308, 1921-1929.

http://dx.doi.org/10.1016/j.disc.2007.04.043

[11] Guo, J. and Wang, K.S. (2010) Posets Associated with Subspaces in a d-Bounded Distance-Regular Graph. Discrete Mathematics, 310, 714-719. http://dx.doi.org/10.1016/j.disc.2009.08.014

[12] Brouwer, A.E., Cohen, A.M. and Neumaier, A. (1989) Distance-Regular Graphs. Springer-Verlag, New York.

http://dx.doi.org/10.1007/978-3-642-74341-2

[13] Wan, Z. (2002) Geometry of Classical Groups over Finite Fields. 2nd Edition, Science Press, Beijing/New York.

[1] Aigner, M. (1979) Combinatorial Theory. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-1-4615-6666-3

[2] Wan, Z. and Huo, Y. (2004) Lattices Generated by Transitive Sets of Subspaces Under Finite Classical Groups. 2nd Edition, Science Press, Beijing.

[3] Liu, Y.H. (2011) Special Lattice of Rough Algebras. Applied Mathematics, 2, 1522-1524.

http://dx.doi.org/10.4236/am.2011.212215

[4] Huo, Y.J. and Wan, Z.-X. (2001) On the Geomericity of Lattices Generated by Orbits of Subspaces under Finite Classical Groups. Journal of Algebra, 243, 339-359.

http://dx.doi.org/10.1006/jabr.2001.8819

[5] Gao, Y. and You, H. (2003) Lattices Generated by Orbits of Subspaces under Finite Singular Classical Groups and Its Characteristic Polynomials. Communications in Algebra, 31, 2927-2950. http://dx.doi.org/10.1081/AGB-120021900

[6] Wang, K.S. and Guo, J. (2009) Lattices Generated by Two Orbits of Subspaces Under Finite Classical Groups. Finite Fields and Their Applications, 15, 236-245. http://dx.doi.org/10.1016/j.ffa.2008.12.008

[7] Wang, K.S. and Li, Z.T. (2008) Lattices Associated with Vector Spaces Over a Finite Field. Linear Algebra and Its Applications, 429, 439-446. http://dx.doi.org/10.1016/j.laa.2008.02.035

[8] Guo, J., Li, Z.T. and Wang, K.S. (2009) Lattices Associated with Totally Isotropic Subspaces in Classical Spaces. Linear Algebra and Its Applications, 431, 1088-1095.

http://dx.doi.org/10.1016/j.laa.2009.04.009

[9] Gao, S.G., Guo, J. and Liu, W. (2007) Lattices Generated by Strongly Closed Subgraphs in d-Bounded Distance-Regular Graphs. European Journal of Combinatorics, 28, 1800-1813.

http://dx.doi.org/10.1016/j.ejc.2006.05.011

[10] Guo, J. and Gao, S.G. (2008) Lattices Generated by Join of Strongly Closed Subgraphs in d-Bounded Distance-Regular Graphs. Discrete Mathematics, 308, 1921-1929.

http://dx.doi.org/10.1016/j.disc.2007.04.043

[11] Guo, J. and Wang, K.S. (2010) Posets Associated with Subspaces in a d-Bounded Distance-Regular Graph. Discrete Mathematics, 310, 714-719. http://dx.doi.org/10.1016/j.disc.2009.08.014

[12] Brouwer, A.E., Cohen, A.M. and Neumaier, A. (1989) Distance-Regular Graphs. Springer-Verlag, New York.

http://dx.doi.org/10.1007/978-3-642-74341-2

[13] Wan, Z. (2002) Geometry of Classical Groups over Finite Fields. 2nd Edition, Science Press, Beijing/New York.