On orthomorphism elements in ordered algebra


TURAN B. , Gurkok H.

TURKISH JOURNAL OF MATHEMATICS, vol.44, no.2, pp.403-408, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 44 Issue: 2
  • Publication Date: 2020
  • Doi Number: 10.3906/mat-1911-28
  • Title of Journal : TURKISH JOURNAL OF MATHEMATICS
  • Page Numbers: pp.403-408

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 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 B-e 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.