DOI QR코드

DOI QR Code

PATH-CONNECTED AND NON PATH-CONNECTED ORTHOMODULAR LATTICES

  • Park, Eun-Soon ;
  • Song, Won-Hee
  • Published : 2009.09.30

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

orthomodular lattice;with finite sites;path-connected;non pathconnected;Boolean algebra

References

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