EMBEDDING ORTHOMORPHISMS d-ALGEBRA IN BIORTHOMORPHISMS AS ORDERED IDEAL

Authors : İbrahim GÖKCAN

Pages : 43-50

Doi:10.54286/ikjm.1025598

View : 21 | Download : 7

Publication Date : 2021-10-10

Article Type : Research Paper

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 Orthinsert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(T_1 *_e T_2);insert ignore into journalissuearticles values(x,y);=T_1 insert ignore into journalissuearticles values(x,T_2 insert ignore into journalissuearticles values(e,y);); for e∈X^+, ∀x,y∈X ve T_1,T_2∈Orthinsert ignore into journalissuearticles values(X,X);. [8] showed that if orthomorphisms are semiprime Dedekind complete f-algebras, it is an ordered ideal in biorthomorphisms. [6] developed an alternative proof for this situation. If X Archimedean Riesz space, Orthinsert ignore into journalissuearticles values(X); is an f-algebra according to compound operation with unit element. [10] showed that if the orthomorphism X is a semi-prime f-algebra, it is a d-algebra. In this study, we investigated embedding orthomorphism in biorthomorphisms when X is uniformly complete d-algebra.
Keywords : d Algebra, Biorthomophism, Ordered ideal, Orthomorphism