Details
| Original language | English |
|---|---|
| Pages (from-to) | 259-287 |
| Number of pages | 29 |
| Journal | Mathematica Bohemica |
| Volume | 138 |
| Issue number | 3 |
| Publication status | Published - 2013 |
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 OX 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 OX 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.
Keywords
- Completely distributive lattice, Continuous function, Continuous lattice, Scott topology, Subset system, Z-continuous, Z-distributive
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Mathematica Bohemica, Vol. 138, No. 3, 2013, p. 259-287.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Z-distributive function lattices
AU - Erné, Marcel
PY - 2013
Y1 - 2013
N2 - 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 OX 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 OX 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.
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 OX 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 OX 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.
KW - Completely distributive lattice
KW - Continuous function
KW - Continuous lattice
KW - Scott topology
KW - Subset system
KW - Z-continuous
KW - Z-distributive
UR - http://www.scopus.com/inward/record.url?scp=84897941867&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:84897941867
VL - 138
SP - 259
EP - 287
JO - Mathematica Bohemica
JF - Mathematica Bohemica
SN - 0862-7959
IS - 3
ER -