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Orthogonality relations on Abelian groups

Published online by Cambridge University Press:  09 April 2009

G. Davis
G. Davis
Affiliation:
La Trobe UniversityBundoora 3083, Australia
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Abstract: An orthogonality relation is an abstract relation on a group having properties similar to the relation on a l-group given by x ⊥ y if ¦x¦ ∧¦y│ = 0. A group G with an orthogonality relation ⊥ is isomorphically represented as a subgroups of the group Γ of continuous global sections of a sheaf of groups. If the stalks of the sheaf are torsion-free and G has and element 1 satisfying 1 ⊥ = (0) then Γ can be ordered so that is is an l-group and x ⊥ y if and only if │x¦ ∧ ¦y¦ = 0 in Γ. An l-group G is complemented if for all x, y ∈ G there is an a ∈ x⊥ ⊕ x⊥⊥ with y ∈ a⊥⊥: equivalent conditions are given for G to be complemented.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 2 , March 1975 , pp. 173 - 179
Copyright
Copyright © Australian Mathematical Society 1975

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