Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-15T05:54:40.461Z Has data issue: false hasContentIssue false
  • English
  • Français

Matrices over orthomodular lattices

Published online by Cambridge University Press:  18 May 2009

J. H. Bevis
J. H. Bevis
Affiliation:
Virginia Polytechnic Institute, Blacksburg, Virginia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper elementary properties are established for matrices whose coordinates are elements of a lattice L. In particular, many of the results of Luce [4] are extended to the case where L is an orthomodular lattice, a lattice with an orthocomplementation denoted by in which a ≦ b ⇒ a ∨(a′ ∧ b) = b. Originally, these were called orthocomplemented weakly modular lattices, Foulis [2]. In Theorem 1 a characterization is given of the nucleus with respect to matrix multiplication, which is in general nonassociative. Matrices with A-1 = transpose (A) are characterized in Lemma 8. Theorems 3 and 4 respectively, give partial characterizations of zero divisors and inverses.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 10 , Issue 1 , January 1969 , pp. 55 - 59
Copyright
Copyright © Glasgow Mathematical Journal Trust 1969

References

REFERENCES

1.Blyth, T. S., ∧-distributive Boolean matrices, Proc. Glasgow Math. Assoc. 7 (1965), 93–100.CrossRefGoogle Scholar
2.Foulis, D. J., Baer *-semigroups, Proc. Amer. Math. Soc. 11 (1960), 648–654.Google Scholar
3.Foulis, D. J., A note on orthomodular lattices, Portugal. Math. 21 (1962), 65–72.Google Scholar
4.Luce, R. D., A note on Boolean matrix theory, Proc. Amer. Math. Soc. 3 (1952), 382–388.CrossRefGoogle Scholar
5.Rutherford, D. E., Inverses of Boolean matrices, Proc. Glasgow Math. Assoc. 6 (1963), 49–53.CrossRefGoogle Scholar