| Title:
|
Lattices of Scott-closed sets (English) |
| Author:
|
Ho, Weng Kin |
| Author:
|
Zhao, Dongsheng |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
2 |
| Year:
|
2009 |
| Pages:
|
297-314 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
domain |
| Keyword:
|
complete semilattice |
| Keyword:
|
Scott-closed set |
| Keyword:
|
C-continuous lattice |
| Keyword:
|
C-algebraic lattice |
| MSC:
|
06A06 |
| MSC:
|
06B23 |
| MSC:
|
06B35 |
| MSC:
|
06D10 |
| MSC:
|
06D99 |
| idZBL:
|
Zbl 1212.06010 |
| idMR:
|
MR2537838 |
| . |
| Date available:
|
2009-08-18T12:25:24Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133435 |
| . |
| Reference:
|
[1] Abramsky S.: Domain Theory and the Logic of Observable Properties.PhD. Thesis, University of London, 1987. MR 1365749 |
| Reference:
|
[2] Abramsky S., Jung A.: Domain Theory.in Handbook of Logic in Computer Science, vol. 3, S. Abramsky, D.M. Gabbay, T.S.E. Maibaum, Eds., Clarendon Press, New York, 1994, pp. 1--168. MR 1365749 |
| Reference:
|
[3] Banaschewski B.: On the topologies of injective spaces.Continuous Lattices and their Applications (Bremen, 1982), Lecture Notes in Pure and Appl. Math., 101, Dekker, New York, 1985, pp. 1--8. Zbl 0614.54033, MR 0825992 |
| Reference:
|
[4] Davey B.A., Priestley H.A.: Introduction to Lattices and Order.second edition, Cambridge Text Books, Cambridge University Press, Cambridge, 1994. Zbl 1002.06001, MR 1902334 |
| Reference:
|
[5] Escardó M.H.: Injective locales over perfect embeddings and algebras of the upper powerlocale monad.Appl. Gen. Topol. 4 (2003), no. 1, 193--200. MR 2021762 |
| Reference:
|
[6] Gierz G., Hoffmann K.H., Keimel K., Lawson J.D., Mislove M.W., Scott D.S.: A Compendium of Continuous Lattices.Springer, Berlin, 1980. MR 0614752 |
| Reference:
|
[7] Gierz G., Hoffmann K.H., Keimel K., Lawson J.D., Mislove M.W., Scott D.S.: Continuous Lattices and Domains.Cambridge University Press, Cambridge, 2003. MR 1975381 |
| Reference:
|
[8] Heckmann R.: Lower and upper power domain constructions commute on all cpos.Inform. Process. Lett. 40 (1991), no. 1, 7--11. Zbl 0748.68038, MR 1134002, 10.1016/S0020-0190(05)80003-1 |
| Reference:
|
[9] W.K. Ho: Theory of Frames.Master Thesis, Nanyang Technological University, 2002. Zbl 1162.06003 |
| Reference:
|
[10] Hoffmann R.E.: Continuous posets, prime spectra of completely distributive lattices, and Hausdorff compactification.in Continuous Lattices, Lecture Notes in Mathematics, 871, Springer, Berlin-Heidelberg, 1981, pp. 159--208. 10.1007/BFb0089907 |
| Reference:
|
[11] Isbell J.R.: Completion of a construction of Johnstone.Proc. Amer. Math. Soc. 85 (1982), 333--334. Zbl 0492.06006, MR 0656096, 10.1090/S0002-9939-1982-0656096-4 |
| Reference:
|
[12] Johnstone P.T.: Scott is not always sober.in Continuous Lattices, Lecture Notes in Mathematics, 871, Springer, Berlin-Heidelberg, 1981, pp. 282--283. Zbl 0469.06002, 10.1007/BFb0089911 |
| Reference:
|
[13] Johnstone P.T.: Stone Spaces.Cambridge Studies in Advanced Mathematics, 3, Cambridge University Press, Cambridge, 1982. Zbl 0586.54001, MR 0698074 |
| Reference:
|
[14] Kock A.: Monads for which structures are adjoint to units.J. Pure Appl. Algebra 104 (1995), no. 1, 41--59. Zbl 0849.18008, MR 1359690, 10.1016/0022-4049(94)00111-U |
| Reference:
|
[15] Lawson J.: The duality of continuous posets.Houston J. Math. 5 (1979), 357--394. Zbl 0428.06003, MR 0559976 |
| Reference:
|
[16] Mac Lane S.: Categories for the Working Mathematician.Springer, New York-Berlin, 1971. Zbl 0906.18001 |
| Reference:
|
[17] Mislove M.W.: Local DCPOs, local CPOs and local completions.Electron. Notes Theor. Comput. Sci., 20, Elsevier, Amsterdam, 1999, pp. 287--300. Zbl 0924.68112, MR 1719008, 10.1016/S1571-0661(04)80085-9 |
| Reference:
|
[18] Papert S.: Which distributive lattices are lattices of closed sets?.Proc. Cambridge Philos. Soc. 55 (1959), 172--176. Zbl 0178.33703, MR 0104601 |
| Reference:
|
[19] Raney G.N.: Completely distributive complete lattices.Proc. Amer. Math. Soc. 3 (1952), 667--680. Zbl 0053.35201, MR 0052392, 10.1090/S0002-9939-1952-0052392-3 |
| Reference:
|
[20] Scott D.: Data types as lattices.SIAM J. Comput. 5 (1976), no. 3, 522--587. Zbl 0337.02018, MR 0437330, 10.1137/0205037 |
| Reference:
|
[21] Schalk A.: Algebras for generalised power constructions.PhD. Thesis, Technische Hochschule Darmstadt, 1993. |
| Reference:
|
[22] Smyth M.B.: Topology.in Handbook of Logic in Computer Science, vol. 1, Oxford University Press, New York, 1992. Zbl 1039.68504, MR 1426367 |
| Reference:
|
[23] Venugopalan G.: Union-complete subset systems.Houston J. Math. 14 (1988), 583--600. Zbl 0689.06005, MR 0998459 |
| Reference:
|
[24] Vickers S.J.: Topology via Logic.Cambridge University Press, Cambridge, 1989. Zbl 0922.54002, MR 1002193 |
| Reference:
|
[25] Zhao D.: N-compactness in L-fuzzy topological spaces.J. Math. Anal. Appl. 128 (1987), 64--79. Zbl 0639.54006, MR 0915967, 10.1016/0022-247X(87)90214-9 |
| Reference:
|
[26] Zhao D.: On projective Z-frames.Canad. Math. Bull. 40 (1997), no. 1, 39--46. Zbl 0871.06007, MR 1443723, 10.4153/CMB-1997-004-4 |
| . |
