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 $C_r$, 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 $I_e$, then $B_e$ = Orthe(C) holds, where Be is the band generated by e in $C_r$. Then, under the same hypotheses, it is obtained that Orthe(C) is an $f$-algebra with a unit $e$.
TURAN, BAHRİ and GÜRKÖK, HÜMA
"On orthomorphism elements in ordered algebra,"
Turkish Journal of Mathematics: Vol. 44:
2, Article 5.
Available at: https://journals.tubitak.gov.tr/math/vol44/iss2/5