2 research outputs found
Weakly partitive families on infinite sets
Given a finite or infinite set and a positive integer , a
{\em binary structure} of base and of rank is a function
.
A subset of is an interval of if for and , and .
The family of intervals of satisfies the following:
, and , where
, are intervals of ;
for every family of intervals of , the intersection of all the elements of is an interval of ; given intervals and of , if , then is an interval of ;
given intervals and of , if , then is an interval of ;
for every up-directed family of intervals of , the union of all the elements of is an interval of .
Given a set , a family of subsets of is weakly partitive if it satisfies the properties above.
After suitably characterizing the elements of a weakly partitive family, we propose a new approach to establish the following \cite{I91}: Given a weakly partitive family on a set , there is a binary structure of base and of rank whose intervals are exactly the elements of
Weakly partitive families on infinite sets
Given a finite or infinite set and a positive integer , a
{\em binary structure} of base and of rank is a function
.
A subset of is an interval of if for and , and .
The family of intervals of satisfies the following:
, and , where
, are intervals of ;
for every family of intervals of , the intersection of all the elements of is an interval of ; given intervals and of , if , then is an interval of ;
given intervals and of , if , then is an interval of ;
for every up-directed family of intervals of , the union of all the elements of is an interval of .
Given a set , a family of subsets of is weakly partitive if it satisfies the properties above.
After suitably characterizing the elements of a weakly partitive family, we propose a new approach to establish the following \cite{I91}: Given a weakly partitive family on a set , there is a binary structure of base and of rank whose intervals are exactly the elements of