Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

PATH-CONNECTED AND NON PATH-CONNECTED ORTHOMODULAR LATTICES

  • Park, Eun-Soon (DEPARTMENT OF MATHEMATICS SOONGSIL UNIVERSITY) ;
  • Song, Won-Hee (DEPARTMENT OF MATHEMATICS GRADUATE SCHOOL SOONGSIL UNIVERSITY)
  • Published : 2009.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

A block of an orthomodular lattice L is a maximal Boolean subalgebra of L. A site is a subalgebra of an orthomodular lattice L of the form S = A $\cap$ B, where A and B are distinct blocks of L. An orthomodular lattice L is called with finite sites if |A $\cap$ B| < $\infty$ for all distinct blocks A, B of L. We prove that there exists a weakly path-connected orthomodular lattice with finite sites which is not path-connected and if L is an orthomodular lattice such that the height of the join-semilattice [ComL]$\vee$ generated by the commutators of L is finite, then L is pathconnected.

Keywords

References

  1. G. Bruns, Block-finite orthomodular lattices, Canad. J. Math. 31 (1979), no. 5, 961–985
  2. G. Bruns and R. Greechie, Blocks and commutators in orthomodular lattices, Algebra Universalis 27 (1990), no. 1, 1–9 https://doi.org/10.1007/BF01190249
  3. J. Dacey, Orthomodular spaces, University of Massachusetts, Ph. D. thesis, 1968
  4. R. Greechie, On the structure of orthomodular lattices satisfying the chain condition, J. Combinatorial Theory 4 (1968), 210–218 https://doi.org/10.1016/S0021-9800(68)80002-X
  5. R. Greechie and L. Herman, Commutator-finite orthomodular lattices, Order 1 (1985), no. 3, 277–284 https://doi.org/10.1007/BF00383604
  6. G. Kalmbach, Orthomodular Lattices, Academic Press, London, 1983
  7. E. Park, Relatively path-connected orthomodular lattices, Bull. Korean Math. Soc. 31 (1994), no. 1, 61–72
  8. E. Park, A note on finite conditions of orthomodular lattices, Commun. Korean Math. Soc. 14 (1999), no. 1, 31–37
  9. M. Roddy, An orthomodular analogue of the Birkhoff-Menger theorem, Algebra Universalis 19 (1984), no. 1, 55–60 https://doi.org/10.1007/BF01191492