EMBEDDING ORTHOMORPHISMS d-ALGEBRA IN BIORTHOMORPHISMS AS ORDERED IDEAL

Abstract

In the historical development of Riesz spaces, we can trace the history of ordered vector spaces to the International Mathematical Congress in Bologna in 1928. Studies related with 𝑓 −Algebras for the Dedekind complete ordered vector space defined in Riesz spaces were initiated by Nakano and their current definition was made by Amemiya, Birkhoff and Pierce.The revival of 𝑓 −algebras, which had a tendency to slow down for a period of time, emerged as a result of Pagter's doctoral thesis [9] and the examination of Alkansas lecture notes by Luxemburg. The concepts of homomorphism, isomorphism, orthomorphism and biorthomorphism in Riesz spaces are defined by Zaanen, Huijsmans, Boulabiar, Buskes and Triki. Algebraic structure of biorthomorphisms defined on Riesz space examined by [8]. 𝑓-Algebra on 𝑂𝑟𝑡ℎ(𝑋, 𝑋) were studied by [8] and [6]. [6] demonstrated that biorthomorphisms space have an 𝑓 −algebraic structure with the help of the product defined as (𝑇 1 * e 𝑇 2 )(𝑥, 𝑦) = 𝑇 1 (𝑥, 𝑇 2 (𝑒, 𝑦)) for 𝑒 ∈ 𝑋 + , ∀𝑥, 𝑦 ∈ 𝑋 and 𝑇 1 , 𝑇 2 ∈ 𝑂𝑟𝑡ℎ(𝑋, 𝑋). [8] showed that if 𝑋 are semiprime Dedekind complete 𝑓 −algebras, 𝑂𝑟𝑡ℎ(𝑋) is an ordered ideal in biorthomorphisms. [6] developed an alternative proof for this situation. If 𝑋 Archimedean Riesz space, 𝑂𝑟𝑡ℎ(𝑋) is an 𝑓 −algebra according to compound operation with unit element. [11] showed that if 𝑋 is a semiprime 𝑓 −algebra, it is a 𝑑 −algebra. In this study, we investigated embedding orthomorphism in biorthomorphisms when 𝑋 is uniformly complete 𝑑 −algebra.