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Extensions of orthomorphisms

Abstract We consider, on an Archimedean Riesz space, the spaces of all linear operators lying between two multiples of the identity (for the order), those leaving all ideals invariant and the order bounded orthomorphisms. We find, if E is uniformly complete, necessary and sufficient conditions for all such operators defined on sublattices of E to extend to the whole of E. Examples are given to show the role of uniform completeness. For the space of all orthomorphisms we give a sufficient condition on E for such an extension to exist.