On orthomorphism elements in ordered algebra

Abstract

Let C be an ordered algebra with a unit e. The class of orthomorphism elements Orthe C of C was introduced and studied by Alekhno in ”The order continuity in ordered algebras”. If C = L G , where G is a Dedekind complete Riesz space, this class coincides with the band Orth G of all orthomorphism operators on G. In this study, the properties of orthomorphism elements similar to properties of orthomorphism operators are obtained. Firstly, it is shown that if C is an ordered algebra such that Cr , the set of all regular elements of C , is a Riesz space with the principal projection property and Orthe C is topologically full with respect to Ie , then Be = Orthe C holds, where Be is the band generated by e in Cr . Then, under the same hypotheses, it is obtained that Orthe C is an f -algebra with a unit e.