A Note on Bi-orthomorphisms

Buskes, Gerard; Page, Robert G.; Yılmaz, Ruşen

doi:10.1007/978-3-0346-0211-2_9

Cited by 6 publications

(7 citation statements)

References 6 publications

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“…Note. 𝜌: 𝑂𝑟𝑡ℎ(𝑋) → 𝑂𝑟𝑡ℎ(𝑋, 𝑋) defined with 𝜌(𝑇)(𝑥, 𝑦) = 𝑇(𝑥𝑦) = 𝑇(𝑥)𝑦 is one-to-one Riesz homomorphism for ∀𝑇 ∈ 𝑂𝑟𝑡ℎ(𝑋) and (𝑥, 𝑦) ∈ 𝑋 × 𝑋 [8]. i.…”

Section: Introductionmentioning

confidence: 99%

“…ii. If 𝑋 is a semiprime Dedekind complete 𝑓 −algebra, 𝑂𝑟𝑡ℎ(𝑋) is an ordered ideal in 𝑂𝑟𝑡ℎ(𝑋, 𝑋) [8]. Theorem 2.…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

mentioning

confidence: 99%

“…[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.…”

mentioning

confidence: 99%

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EMBEDDING ORTHOMORPHISMS d-ALGEBRA IN BIORTHOMORPHISMS AS ORDERED IDEAL

Gökcan

2021

Ikonion Journal of Mathematics

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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.

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Section: Introductionmentioning

confidence: 99%

“…ii. If 𝑋 is a semiprime Dedekind complete 𝑓 −algebra, 𝑂𝑟𝑡ℎ(𝑋) is an ordered ideal in 𝑂𝑟𝑡ℎ(𝑋, 𝑋) [8]. Theorem 2.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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EMBEDDING ORTHOMORPHISMS d-ALGEBRA IN BIORTHOMORPHISMS AS ORDERED IDEAL

Gökcan

2021

Ikonion Journal of Mathematics

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“…(2) a bi-orthomorphism if it is a separately order bounded bilinear map such that x^y = 0 in A implies T (z; x)^y = 0 for all z 2 A + , when A = B (…rst appears a paper by G. Buskes, R. Page Jr and R. Yilmaz in [10] in 2009). The following theorem is obvious from the above de…nitions.…”

Section: Introductionmentioning

confidence: 99%