A dichotomy result for locally compact sober dcpos

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A dichotomy result for locally compact sober dcpos. / Jia, Xiaodong; Jung, Achim; Li, Qingguo.

In: Houston Journal of Mathematics, Vol. 45, No. 3, 30.11.2019, p. 935-951.

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Jia, Xiaodong ; Jung, Achim ; Li, Qingguo. / A dichotomy result for locally compact sober dcpos. In: Houston Journal of Mathematics. 2019 ; Vol. 45, No. 3. pp. 935-951.

Bibtex

@article{29552f7157af4267aed94119d4a874a8,
title = "A dichotomy result for locally compact sober dcpos",
abstract = "The second author proved in [7] that each cartesian closed categoryof pointed domains and Scott-continuous functions is contained in eitherthe category of Lawson-compact domains or that of L-domains, and thisresult eventually led to a classification of continuous domains with respectto cartesian closedness, as laid out in [8].In this paper, we generalise this result to the category LcS of pointedlocally compact sober dcpos and Scott-continuous functions, and showthat any cartesian closed full subcategory of LcS is contained in eitherthe category of stably compact dcpos or that of L-dcpos. (Note thatfor domains Lawson-compactness and stable compactness are equivalent.)As we will show, this entails that any candidate for solving the Jung-Tixproblem in LcS must be stably compact.To prove our dichotomy result, we first show that any dcpo with a core-compact function space must be meet-continuous; then we prove that afunction space in LcS is meet-continuous only if either its input dcpo iscoherent or its output dcpo has complete principal ideals.",
author = "Xiaodong Jia and Achim Jung and Qingguo Li",
year = "2019",
month = nov,
day = "30",
language = "English",
volume = "45",
pages = "935--951",
journal = "Houston Journal of Mathematics",
issn = "0362-1588",
publisher = "University of Houston",
number = "3",

}

RIS

TY - JOUR

T1 - A dichotomy result for locally compact sober dcpos

AU - Jia, Xiaodong

AU - Jung, Achim

AU - Li, Qingguo

PY - 2019/11/30

Y1 - 2019/11/30

N2 - The second author proved in [7] that each cartesian closed categoryof pointed domains and Scott-continuous functions is contained in eitherthe category of Lawson-compact domains or that of L-domains, and thisresult eventually led to a classification of continuous domains with respectto cartesian closedness, as laid out in [8].In this paper, we generalise this result to the category LcS of pointedlocally compact sober dcpos and Scott-continuous functions, and showthat any cartesian closed full subcategory of LcS is contained in eitherthe category of stably compact dcpos or that of L-dcpos. (Note thatfor domains Lawson-compactness and stable compactness are equivalent.)As we will show, this entails that any candidate for solving the Jung-Tixproblem in LcS must be stably compact.To prove our dichotomy result, we first show that any dcpo with a core-compact function space must be meet-continuous; then we prove that afunction space in LcS is meet-continuous only if either its input dcpo iscoherent or its output dcpo has complete principal ideals.

AB - The second author proved in [7] that each cartesian closed categoryof pointed domains and Scott-continuous functions is contained in eitherthe category of Lawson-compact domains or that of L-domains, and thisresult eventually led to a classification of continuous domains with respectto cartesian closedness, as laid out in [8].In this paper, we generalise this result to the category LcS of pointedlocally compact sober dcpos and Scott-continuous functions, and showthat any cartesian closed full subcategory of LcS is contained in eitherthe category of stably compact dcpos or that of L-dcpos. (Note thatfor domains Lawson-compactness and stable compactness are equivalent.)As we will show, this entails that any candidate for solving the Jung-Tixproblem in LcS must be stably compact.To prove our dichotomy result, we first show that any dcpo with a core-compact function space must be meet-continuous; then we prove that afunction space in LcS is meet-continuous only if either its input dcpo iscoherent or its output dcpo has complete principal ideals.

M3 - Article

VL - 45

SP - 935

EP - 951

JO - Houston Journal of Mathematics

JF - Houston Journal of Mathematics

SN - 0362-1588

IS - 3

ER -