author = "Gutman A.E.",

title = "Locally one-dimensional K-spaces and $\sigma$-distributive Boolean algebras",

journal = "Siberian Adv. Math.",

year = "1995",

volume = "5",

number = "1",

pages = "42--48",

annote = "It is known that all band preserving operators acting in a universally complete K-space are regular if and only if the K-space is locally one-dimensional. In addition, a K-space with base $B$ is locally one-dimensional if and only if $R^{\land}=R$ in $V^{(B)}$. It seems to have been unknown so far whether there exist nondiscrete locally one-dimensional K-spaces. In the present note we give a positive answer to the question. As an auxiliary result, we establish that a K-space is locally one-dimensional if and only if its base is $\sigma$-distributive.",

keywords = "locally one-dimensional K-space, discrete K-space, $\sigma$-distributive Boolean algebra, $\sigma$-inductive Boolean algebra, atomic Boolean algebra, regular operator, real numbers in a Boolean-valued universe"

author = "Gutman A.E.",

title = "Locally one-dimensional K-spaces and $\sigma$-distributive Boolean algebras",

booktitle = "Siberian Conference on Applied and Industrial Mathematics (Novosibirsk, July 25--29, 1994): Proceedings",

address = "Novosibirsk",

publisher = "IM SB RAS",

year = "1997",

volume = "1",

pages = "103--108",

language = "russian",

annote = "It is known that all band preserving operators acting in a universally complete K-space are regular if and only if the K-space is locally one-dimensional. In addition, a K-space with base $B$ is locally one-dimensional if and only if $R^{\land}=R$ in $V^{(B)}$. It seems to have been unknown so far whether there exist nondiscrete locally one-dimensional K-spaces. In the present note we give a positive answer to the question. As an auxiliary result, we establish that a K-space is locally one-dimensional if and only if its base is $\sigma$-distributive."

author = "Gutman A.E.",

title = "Locally one-dimensional K-spaces",

journal = "Proc. Acad. Sci.",

year = "1997",

volume = "353",

number = "5",

pages = "590--591",

language = "russian",

annote = "It is known that all band preserving operators acting in a universally complete K-space are regular if and only if the K-space is locally one-dimensional. In addition, a K-space with base $B$ is locally one-dimensional if and only if $R^{\land}=R$ in $V^{(B)}$. It seems to have been unknown so far whether there exist nondiscrete locally one-dimensional K-spaces. In the present note we give a positive answer to the question. As an auxiliary result, we establish that a K-space is locally one-dimensional if and only if its base is $\sigma$-distributive."

author = "Gutman A.E.",

title = "Locally one-dimensional complete vector lattices",

journal = "Doklady Math.",

year = "1997",

volume = "55",

number = "2",

pages = "240--241",

annote = "It is known that all band preserving operators acting in a universally complete K-space are regular if and only if the K-space is locally one-dimensional. In addition, a K-space with base $B$ is locally one-dimensional if and only if $R^{\land}=R$ in $V^{(B)}$. It seems to have been unknown so far whether there exist nondiscrete locally one-dimensional K-spaces. In the present note we give a positive answer to the question. As an auxiliary result, we establish that a K-space is locally one-dimensional if and only if its base is $\sigma$-distributive."

author = "Gutman A.E. and Kusraev A.G. and Kutateladze S.S.",

note = "Preprint / IAMI VSC RAS; N 3",

title = "The Wickstead problem",

address = "Vladikavkaz",

year = "2007",

pages = "44",

doi = "10.13140/RG.2.2.22573.79849",

annote = "In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice.",

keywords = "band preserving operator, universally complete vector lattice, $\sigma$-distributive Boolean algebra, local Hamel basis, transcendence basis, derivation, Boolean valued representation"

author = "Gutman A.E. and Kusraev A.G. and Kutateladze S.S.",

note = "Electronic preprint / arXiv:0712.2378 [math.FA]",

howpublished = "Electronic",

title = "The Wickstead problem",

address = "arXiv.org",

year = "2007",

pages = "44",

annote = "In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice.",

keywords = "band preserving operator, universally complete vector lattice, $\sigma$-distributive Boolean algebra, local Hamel basis, transcendence basis, derivation, Boolean valued representation"

author = "Gutman A.E. and Kusraev A.G. and Kutateladze S.S.",

howpublished = "Electronic",

title = "The Wickstead problem",

journal = "Sib. Electron. Math. Rep.",

year = "2008",

volume = "5",

pages = "293--333",

annote = "In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice.",

keywords = "band preserving operator, universally complete vector lattice, $\sigma$-distributive Boolean algebra, local Hamel basis, transcendence basis, derivation, Boolean valued representation"