Orthogonality relations on Abelian groups

Abstract

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.