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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
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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

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