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 wasintroduced and studied by Alekhno in ”The order continuity in ordered algebras”. If C = L(G), where G is a Dedekindcomplete 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 isshown that if C is an ordered algebra such that Cr , the set of all regular elements of C , is a Riesz space with theprincipal projection property and Orthe(C) is topologically full with respect to Ie , then Be = Orthe(C) holds, whereBe is the band generated by e in Cr . Then, under the same hypotheses, it is obtained that Orthe(C) is an f -algebrawith a unit e.