Abstract
Suda (2012) extended the Erdős-Ko-Rado theorem to designs in strongly regularized semilattices. In this paper we generalize Suda’s results in regularized semilattices and partition regularized semilattices, give many examples for these semilattices and obtain their intersection theorems.
Similar content being viewed by others
References
Borg P. Intersecting and cross-intersecting families of labeled sets. Electron J Combin, 2008, 15: N9
Brouwer A E, Cohen A M, Neumaier A. Distance-Regular Graphs. Berlin Heidelberg: Springer-Verlag, 1989
Cameron P J. Projective and polar spaces. QMW Maths Notes, vol. 13. London: QMW, 1991
Cameron P J, Ku C Y. Intersecting families of permutations. European J Combin, 2003, 24: 881–890
Deza M, Frankl P. On the maximum number of permutations with given maximal or minimal distance. J Combin Theory Ser A, 1977, 22: 352–362
Deza M, Frankl P. Erdős-Ko-Rado theorem—22 years later. SIAM J Algebraic Discrete Methods, 1983, 4: 419–431
Delarte P. Association schemes and t-designs in regular semilattices. J Combin Theory Ser A, 1976, 20: 230–243
Ellis D, Friedgut E, Pilpel H. Intersection families of permutations. J Amer Math Soc, 2011, 24: 649–682
Erdős P, Ko C, Rado R. Intersection theorems for systems of finite sets. Quart J Math Oxford Ser, 1961, 2: 313–318
Frankl P, Wilson R M. The Erdős-Ko-Rado theorem for vector spaces. J Combin Theory Ser A, 1986, 43: 228–236
Fu T. Erdős-Ko-Rado-type results over J q(n, d),H q(n, d) and their designs. Discrete Math, 1999, 196: 137–151
Gao S, Guo J, Liu W. Lattices generated by strongly closed subgraphs in d-bounded distance-regular graphs. European J Combin, 2007, 28: 1800–1813
Gao S, Guo J, Zhang B, Fu L. Subspaces in d-bounded distance-regular graphs and their applications. European J Combin, 2008, 29: 592–600
Gao S, Guo J. The graphs induced by maximal totally isotropic flats of affine-unitary spaces. Finite Fields Appl, 2009, 15: 185–194
Gao Y. Lattices generated by orbits of subspaces under finite singular unitary group and its characteristic polynomials. Linear Algebra Appl, 2003, 368: 243–268
Gao Y, Fu X. Lattices generated by orbits of subspaces under finite singular orthogonal groups I. Finite Fields Appl, 2010, 16: 385–400
Gao Y, You H. Lattices generated by orbits of subspaces under finite singular classical groups and its characteristic polynomials. Comm Algebra, 2003, 31: 2927–2950
Gao Y, Xu J. Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups I. Linear Algebra Appl, 2009, 431: 1455–1476
Gao Y, Xu J. Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups II. Finite Fields Appl, 2009, 15: 360–374
Geng X, Li Y. Erdős-Ko-Rado theorems of labeled sets. Acta Math Appl Sin Engl Ser, 2012, 28: 127–130
Guo J. Suborbits of (m, k)-isotropic subspaces under finite singular classical groups. Finite Fields Appl, 2010, 16: 126–136
Guo J, Gao S. Lattices generated by join of strongly closed subgraphs in d-bounded distance-regular graphs. Discrete Math, 2008, 308: 1921–1929
Guo J, Gao S, Wang K. Lattices generated by subspaces in d-bounded distance-regular graphs. Discrete Math, 2008, 308: 5260–5264
Guo J, Gao S. A generalization of dual polar graph of orthogonal space. Finite Fields Appl, 2009, 15: 661–672
Guo J, Gao S. Character tables of the association schemes obtained from the finite affine classical groups acting on the sets of maximal totally isotropic flats. Adv Geometry, 2011, 11: 303–311
Guo J, Li Z, Wang K. Lattices associated with totally isotropic subspaces in classical spaces. Linear Algebra Appl, 2009, 431: 1088–1095
Guo J, Nan J. Lattices generated by orbits of flats under finite affine-symplectic groups. Linear Algebra Appl, 2009, 431: 536–542
Guo J, Wang K. An Erdős-Ko-Rado theorem in general linear groups. ArXiv:1107.3178
Hsieh W N. Intersection theorems for finite vector spaces. Discrete Math, 1975, 12: 1–16
Huang T. An analogue of the Erdős-Ko-Rado theorem for the distance-regular graphs of bilinear forms. Discrete Math, 1987, 64: 191–198
Huo Y, Liu Y, Wan Z. Lattices generated by transitive sets of subspaces under finite classical groups I. Comm Algebra, 1992, 20: 1123–1144
Huo Y, Liu Y, Wan Z. Lattices generated by transitive sets of subspaces under finite classical groups II, the orthogonal case of odd characteristic. Comm Algebra, 1993, 20: 2685–2727
Huo Y, Liu Y, Wan Z. Lattices generated by transitive sets of subspaces under finite classical groups III, the orthogonal case of even characteristic. Comm Algebra, 1993, 21: 2351–2393
Huo Y, Wan Z. Lattices generated by transitive sets of subspaces under finite pseudo-symplectic groups. Comm Algebra, 1995, 23: 3757–3777
Huo Y, Wan Z. On the geometricity of lattices generated by orbits of subspaces under finite classical groups. J Algebra, 2001, 243: 339–359
Ku C Y, Leader I. An Erdős-Ko-Rado theorem for partial permutations. Discrete Math, 2006, 306: 74–86
Li Y, Wang J. Erdős-Ko-Rado-type theorems for colored sets. Electron J Combin, 2007, 14: R1
Nan J, Guo J. Lattices generated by two orbits of subspaces under finite singular classical groups. Comm Algebra, 2010, 38: 2026–2036
Rands B M I. An extension of the Erdős, Ko, Rado theorem to t-designs. J Combin Theory Ser A, 1982, 32: 391–395
Suzuki H. On strongly closed subgraphs of highly regular graphs. European J Combin, 1995, 16: 197–220
Suda S. A generalization of the Erdős-Ko-Rado theorem to t-designs in certain semilattices. Discrete Math, 2012, 312: 1827–1831
Tanaka H. Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs. J Combin Theory Ser A, 2006, 113: 903–910
Terwilliger P. The incidence algebra of a uniform poset. In: Coding Theory and Design Theory part I: Coding Theory. IMA volumes in Mathematics and its Applications, vol. 20. New York: Springer, 1990, 193–212
Vanhove F, Pepe V, Storme L. Theorems of Erdős-Ko-Rado-type in polar spaces. J Combin Theory Ser A, 2011, 118: 1291–1312
Wan Z. Geometry of Classical Groups over Finite Fields, 2nd edition. Beijing/New York: Science Press, 2002
Wang K, Feng Y. Lattices generated by orbits of flats under affine groups. Comm Algebra, 34, 2006: 1691-1697
Wang K, Guo J, Li F. Singular linear space and its application. Finite Fields Appl, 2011, 17: 395–406
Wang K, Guo J. Lattices generated by orbits of totally isotropic flats under finite affine-classical groups. Finite Fields Appl, 2008, 14: 571–578
Wang K, Guo J, Li F. Association schemes based on attenuated spaces. European J Combin, 2010, 31: 297–305
Wang K, Li Z. Lattices associated with vector spaces over a finite field. Linear Algebra Appl, 429, 2008: 439–446
Weng C. D-bounded distance-regular graphs. European J Combin, 1997, 18: 211–229
Weng C. Classical distance-regular graphs of negative type. J Combin Theory Ser B, 1999, 76: 93–116
Wilson R M. The exact bound in the Erdős-Ko-Rado theorem. Combinatorica, 1984, 4: 247–257
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guo, J., Ma, J. & Wang, K. Erdős-Ko-Rado theorems in certain semilattices. Sci. China Math. 56, 2393–2407 (2013). https://doi.org/10.1007/s11425-012-4563-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-012-4563-z
Keywords
- Erdős-Ko-Rado theorem
- semilattice
- regularized semilattice
- strongly regularized semilattice
- partition regularized semilattice
- partition strongly regularized semilattice