𝒵 -distributive function lattices

Marcel Erné

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 3, page 259-287
  • ISSN: 0862-7959

Abstract

top
It is known that for a nonempty topological space X and a nonsingleton complete lattice Y endowed with the Scott topology, the partially ordered set [ X , Y ] of all continuous functions from X into Y is a continuous lattice if and only if both Y and the open set lattice 𝒪 X are continuous lattices. This result extends to certain classes of 𝒵 -distributive lattices, where 𝒵 is a subset system replacing the system 𝒟 of all directed subsets (for which the 𝒟 -distributive complete lattices are just the continuous ones). In particular, it is shown that if [ X , Y ] is a complete lattice then it is supercontinuous (i.e. completely distributive) iff both Y and 𝒪 X are supercontinuous. Moreover, the Scott topology on Y is the only one making that equivalence true for all spaces X with completely distributive topology. On the way to these results, we find necessary and sufficient conditions for [ X , Y ] to be complete, and some new, purely topological characterizations of continuous lattices by continuity conditions on their (infinitary) lattice operations.

How to cite

top

Erné, Marcel. "$\mathcal {Z}$-distributive function lattices." Mathematica Bohemica 138.3 (2013): 259-287. <http://eudml.org/doc/260626>.

@article{Erné2013,
abstract = {It is known that for a nonempty topological space $X$ and a nonsingleton complete lattice $Y$ endowed with the Scott topology, the partially ordered set $[X,Y]$ of all continuous functions from $X$ into $Y$ is a continuous lattice if and only if both $Y$ and the open set lattice $\mathcal \{O\} X$ are continuous lattices. This result extends to certain classes of $\mathcal \{Z\}$-distributive lattices, where $\mathcal \{Z\}$ is a subset system replacing the system $\mathcal \{D\}$ of all directed subsets (for which the $\mathcal \{D\}$-distributive complete lattices are just the continuous ones). In particular, it is shown that if $[X,Y]$ is a complete lattice then it is supercontinuous (i.e. completely distributive) iff both $Y$ and $\mathcal \{O\} X$ are supercontinuous. Moreover, the Scott topology on $Y$ is the only one making that equivalence true for all spaces $X$ with completely distributive topology. On the way to these results, we find necessary and sufficient conditions for $[X,Y]$ to be complete, and some new, purely topological characterizations of continuous lattices by continuity conditions on their (infinitary) lattice operations.},
author = {Erné, Marcel},
journal = {Mathematica Bohemica},
keywords = {completely distributive lattice; continuous function; continuous lattice; Scott topology; subset system; $\mathcal \{Z\}$-continuous; $\mathcal \{Z\}$-distributive; completely distributive lattice; continuous lattice; Scott topology; subset system; -continuous; -distributive},
language = {eng},
number = {3},
pages = {259-287},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$\mathcal \{Z\}$-distributive function lattices},
url = {http://eudml.org/doc/260626},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Erné, Marcel
TI - $\mathcal {Z}$-distributive function lattices
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 3
SP - 259
EP - 287
AB - It is known that for a nonempty topological space $X$ and a nonsingleton complete lattice $Y$ endowed with the Scott topology, the partially ordered set $[X,Y]$ of all continuous functions from $X$ into $Y$ is a continuous lattice if and only if both $Y$ and the open set lattice $\mathcal {O} X$ are continuous lattices. This result extends to certain classes of $\mathcal {Z}$-distributive lattices, where $\mathcal {Z}$ is a subset system replacing the system $\mathcal {D}$ of all directed subsets (for which the $\mathcal {D}$-distributive complete lattices are just the continuous ones). In particular, it is shown that if $[X,Y]$ is a complete lattice then it is supercontinuous (i.e. completely distributive) iff both $Y$ and $\mathcal {O} X$ are supercontinuous. Moreover, the Scott topology on $Y$ is the only one making that equivalence true for all spaces $X$ with completely distributive topology. On the way to these results, we find necessary and sufficient conditions for $[X,Y]$ to be complete, and some new, purely topological characterizations of continuous lattices by continuity conditions on their (infinitary) lattice operations.
LA - eng
KW - completely distributive lattice; continuous function; continuous lattice; Scott topology; subset system; $\mathcal {Z}$-continuous; $\mathcal {Z}$-distributive; completely distributive lattice; continuous lattice; Scott topology; subset system; -continuous; -distributive
UR - http://eudml.org/doc/260626
ER -

References

top
  1. Bandelt, H.-J., Erné, M., 10.1016/0022-4049(83)90057-9, J. Pure Appl. Algebra 30 (1983), 219-226. (1983) Zbl0523.06001MR0724033DOI10.1016/0022-4049(83)90057-9
  2. Bandelt, H.-J., Erné, M., Representations and embeddings of M -distributive lattices, Houston J. Math. 10 (1984), 315-324. (1984) Zbl0551.06014MR0763234
  3. Baranga, A., 10.1016/0012-365X(94)00307-5, Discrete Math. 152 (1996), 33-45. (1996) Zbl0851.06003MR1388630DOI10.1016/0012-365X(94)00307-5
  4. Baranga, A., Z-continuous posets, topological aspects, Stud. Cercet. Mat. 49 (1997), 3-16. (1997) Zbl0883.06007MR1671509
  5. Erné, M., 10.1007/BFb0089904, Continuous Lattices. Proc. Conf., Bremen 1979 Lect. Notes Math. 871 61-96 (1981), B. Banaschewski, R.-E. Hoffmann Springer, Berlin. (1981) DOI10.1007/BFb0089904
  6. Erné, M., Adjunctions and standard constructions for partially ordered sets, Contributions to General Algebra. Proc. Klagenfurt Conf. 1982 Contrib. Gen. Algebra 2 Hölder, Wien 77-106 (1983), G. Eigenthaler et al. Contributions to General Algebra. (1983) Zbl0533.06001MR0721648
  7. Erné, M., The ABC of order and topology, Category Theory at Work. Proc. Workshop, Bremen 1991 Res. Expo. Math. 18 57-83 (1991), H. Herrlich, H.-E. Porst Heldermann, Berlin. (1991) Zbl0735.18005MR1147919
  8. Erné, M., Algebraic ordered sets and their generalizations, I. Rosenberg Algebras and Orders. Kluwer Academic Publishers. NATO ASI Ser. C, Math. Phys. Sci. 389 Kluwer Acad. Publ., Dordrecht 113-192 (1993). (1993) Zbl0791.06007MR1233790
  9. Erné, M., 10.1023/A:1008657800278, Appl. Categ. Struct. 7 (1999), 31-70. (1999) Zbl0939.06005MR1714179DOI10.1023/A:1008657800278
  10. Erné, M., Minimal bases, ideal extensions, and basic dualities, Topol. Proc. 29 (2005), 445-489. (2005) Zbl1128.06001MR2244484
  11. Erné, M., Closure, F. Mynard, E. Pearl Beyond Topology. AMS Contemporary Mathematics 486 Providence, R.I. (2009), 163-238. (2009) Zbl1209.08001MR2555999
  12. Erné, M., 10.1016/j.topol.2009.03.029, Topology Appl. 156 (2009), 2054-2069. (2009) Zbl1190.54022MR2532134DOI10.1016/j.topol.2009.03.029
  13. Erné, M., Gatzke, H., Convergence and continuity in partially ordered sets and semilattices, Continuous Lattices and Their Applications. Proc. 3rd Conf., Bremen 1982 Lect. Notes Pure Appl. Math. 101 9-40 (1985). (1985) Zbl0591.54029MR0825993
  14. Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M., Scott, D. S., A Compendium of Continuous Lattices, Springer, Berlin (1980). (1980) Zbl0452.06001MR0614752
  15. Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M., Scott, D. S., Continuous Lattices and Domains, Encyclopedia of Mathematics and Its Applications 93 Cambridge University Press, Cambridge (2003). (2003) Zbl1088.06001MR1975381
  16. Hoffmann, R.-E., 10.1007/BFb0089907, Continuous Lattices. Proc. Conf., Bremen 1979 Lect. Notes Math. 871 159-208 (1981), B. Banaschewski, R.-E. Hoffmann Springer, Berlin. (1981) Zbl0476.06005DOI10.1007/BFb0089907
  17. Isbell, J., 10.1090/S0002-9939-1982-0656096-4, Proc. Am. Math. Soc. 85 (1982), 333-334. (1982) Zbl0492.06006MR0656096DOI10.1090/S0002-9939-1982-0656096-4
  18. Keimel, K., 10.1016/j.entcs.2009.11.025, Electronical Notes in Th. Computer Sci. 257 (2009), 35-54. (2009) DOI10.1016/j.entcs.2009.11.025
  19. Kříž, I., Pultr, A., A spatiality criterion and an example of a quasitopology which is not a topology, Houston J. Math. 15 (1989), 215-234. (1989) Zbl0695.54002MR1022063
  20. Meseguer, J., Order completion monads, Algebra Univers. 16 (1983), 63-82. (1983) Zbl0522.18005MR0690831
  21. Novak, D., 10.1090/S0002-9947-1982-0662058-8, Trans. Am. Math. Soc. 272 (1982), 645-667. (1982) Zbl0504.06003MR0662058DOI10.1090/S0002-9947-1982-0662058-8
  22. Qin, F., Function space of Z-continuous lattices, Fuzzy Syst. Math. 14 (2000), 31-35 Chinese. (2000) MR1802864
  23. Raney, G. N., 10.1090/S0002-9939-1953-0058568-4, Proc. Am. Math. Soc. 4 (1953), 518-522. (1953) Zbl0053.35201MR0058568DOI10.1090/S0002-9939-1953-0058568-4
  24. Scott, D. S., Continuous lattices, Toposes, Algebraic Geometry and Logic. Dalhousie Univ. Halifax 1971, Lect. Notes Math. 274 97-136 (1972), Springer, Berlin. (1972) Zbl0239.54006MR0404073
  25. Venugopalan, G., Z-continuous posets, Houston J. Math. 12 (1986), 275-294. (1986) Zbl0614.06007MR0862043
  26. Wright, J. B., Wagner, E. G., Thatcher, J. W., 10.1016/0304-3975(78)90040-3, Theor. Comput. Sci. 7 (1978), 57-77. (1978) Zbl0732.06001MR0480224DOI10.1016/0304-3975(78)90040-3
  27. Wyler, O., 10.1007/BFb0089920, B. Banaschewski, R.-E. Hoffmann Continuous Lattices. Proc. Conf., Bremen 1979, Lect. Notes Math. 871 384-389 (1981), Springer, Berlin. (1981) Zbl0488.54018DOI10.1007/BFb0089920

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.