Geodesic
Lattices of Scott-closed sets
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 2, pp. 297-314 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $P\mapsto C(P)$ to a left adjoint functor from the category {\bf DCPO} of dcpo's to the category {\bf CPAlg} of C-prealgebraic lattices.
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 {\bf DCPO} of dcpo's to the category {\bf CPAlg} of C-prealgebraic lattices.
Classification : 06A06, 06B23, 06B35, 06D10, 06D99
Keywords: domain; complete semilattice; Scott-closed set; C-continuous lattice; C-algebraic lattice 
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     author = {Ho, Weng Kin and Zhao, Dongsheng},
     title = {Lattices of {Scott-closed} sets},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {297--314},
     year = {2009},
     volume = {50},
     number = {2},
     mrnumber = {2537838},
     zbl = {1212.06010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2009_50_2_a9/}
}
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Ho, Weng Kin; Zhao, Dongsheng. Lattices of Scott-closed sets. Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 2, pp. 297-314. http://geodesic.mathdoc.fr/item/CMUC_2009_50_2_a9/