Lattices of Scott-closed sets

Weng Kin Ho; Dong Sheng Zhao

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 2, page 297-314
  • ISSN: 0010-2628

Abstract

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A dcpo P is continuous if and only if the lattice C ( P ) of all Scott-closed subsets of P is completely distributive. However, in the case where P is a non-continuous dcpo, little is known about the order structure of C ( P ) . In this paper, we study the order-theoretic properties of C ( P ) for general dcpo’s P . The main results are: (i) every C ( P ) is C-continuous; (ii) a complete lattice L is isomorphic to C ( P ) for a complete semilattice P if and only if L is weak-stably C-algebraic; (iii) for any two complete semilattices P and Q , P and Q are isomorphic if and only if C ( P ) and C ( Q ) are isomorphic. In addition, we extend the function P ↦ C ( P ) to a left adjoint functor from the category DCPO of dcpo’s to the category CPAlg of C-prealgebraic lattices.

How to cite

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Ho, Weng Kin, and Zhao, Dong Sheng. "Lattices of Scott-closed sets." Commentationes Mathematicae Universitatis Carolinae 50.2 (2009): 297-314. <http://eudml.org/doc/32500>.

@article{Ho2009,
abstract = {A dcpo $P$ is continuous if and only if the lattice $C(P)$ of all Scott-closed subsets of $P$ is completely distributive. However, in the case where $P$ is a non-continuous dcpo, little is known about the order structure of $C(P)$. In this paper, we study the order-theoretic properties of $C(P)$ for general dcpo’s $P$. The main results are: (i) every $C(P)$ is C-continuous; (ii) a complete lattice $L$ is isomorphic to $C(P)$ for a complete semilattice $P$ if and only if $L$ is weak-stably C-algebraic; (iii) for any two complete semilattices $P$ and $Q$, $P$ and $Q$ are isomorphic if and only if $C(P)$ and $C(Q)$ are isomorphic. In addition, we extend the function $P\mapsto C(P)$ to a left adjoint functor from the category DCPO of dcpo’s to the category CPAlg of C-prealgebraic lattices.},
author = {Ho, Weng Kin, Zhao, Dong Sheng},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {domain; complete semilattice; Scott-closed set; C-continuous lattice; C-algebraic lattice; domain; complete semilattice; Scott-closed set; C-algebraic lattice},
language = {eng},
number = {2},
pages = {297-314},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Lattices of Scott-closed sets},
url = {http://eudml.org/doc/32500},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Ho, Weng Kin
AU - Zhao, Dong Sheng
TI - Lattices of Scott-closed sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 2
SP - 297
EP - 314
AB - A dcpo $P$ is continuous if and only if the lattice $C(P)$ of all Scott-closed subsets of $P$ is completely distributive. However, in the case where $P$ is a non-continuous dcpo, little is known about the order structure of $C(P)$. In this paper, we study the order-theoretic properties of $C(P)$ for general dcpo’s $P$. The main results are: (i) every $C(P)$ is C-continuous; (ii) a complete lattice $L$ is isomorphic to $C(P)$ for a complete semilattice $P$ if and only if $L$ is weak-stably C-algebraic; (iii) for any two complete semilattices $P$ and $Q$, $P$ and $Q$ are isomorphic if and only if $C(P)$ and $C(Q)$ are isomorphic. In addition, we extend the function $P\mapsto C(P)$ to a left adjoint functor from the category DCPO of dcpo’s to the category CPAlg of C-prealgebraic lattices.
LA - eng
KW - domain; complete semilattice; Scott-closed set; C-continuous lattice; C-algebraic lattice; domain; complete semilattice; Scott-closed set; C-algebraic lattice
UR - http://eudml.org/doc/32500
ER -

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