𝒵 -distributive function lattices

Marcel Erné

Mathematica Bohemica (2013)

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

Abstract

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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

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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

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