All Questions
9 questions
5
votes
1
answer
298
views
Partition induced by a cover
Let $X$ be a set and let $(Y_i)_{i \in I}$ be a family of (not necessarily pairwise disjoint) subsets covering $X$,
$$ X = \bigcup_{i\in I} Y_i.$$
For any subset $J \subseteq I$, we then define
$$ Y_J ...
- 9,853
2
votes
1
answer
285
views
Size of antichains in powerset of $\mathbb N$
Take a countably infinite set $S$, say $\mathbb N$. Is it possible for there to be an antichain in $\mathcal P(S)$ (with the inclusion ordering) of continuum cardinality?
- 33
4
votes
1
answer
239
views
Cofinal rectangles in poset
Suppose $(P, <)$ is a poset of cofinality $\aleph_2$ and additivity (least cardinality of an unbounded subset) $\aleph_1$. Can we conclude the existence of a cofinal subset of order-type $\omega_1 \...
- 505
4
votes
1
answer
120
views
Searching for cofinal subsets of directed sets subject to finite constraints
Let $(P,\leq)$ be a directed set with uncountable cofinality. For every element $p\in P$, we are given a finite set $c_p\subset P\smallsetminus \{p\}$ of "incompatible elements". We say that ...
- 380
9
votes
0
answers
250
views
Distributivity of certain infinite products
Suppose we have a sequence of posets $\{\mathbb P_n : n\in\omega\}$ such that for each $n$, $\mathbb P_{n+1}$ is $|\mathbb P_n|^+$-distributive. Is $\prod_{n>0} \mathbb P_n$ necessarily $|\mathbb ...
- 18.6k
1
vote
1
answer
140
views
Thinning directed sets ${\frak P}$ of partitions of $\omega$ with no ${\frak P}$-discrete subsets
This question branches from Taras Banakh's recent question on a cardinal characteristic connected to families of partitions that are directed in the ordering of partition refinement.
A partition $\...
- 48.9k
6
votes
2
answers
1k
views
Poset dimension and width (Dilworth's theorem)
For a given poset $P$, let $\mathrm{dim}(P)$ denote the least cardinal $\kappa$ such that there exists a $\kappa$-sized collection of linear extensions of $P$, say $\mathcal{L}$, such that $\leq_P = \...
- 1,006
7
votes
2
answers
344
views
Surjective order-preserving map $f:{\cal P}(X)\to \text{Part}(X)$
Let $X$ be a set, and let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ ...
- 48.9k
14
votes
3
answers
1k
views
Order homomorphism functions on $\omega_1$
I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains incomplete, so I thought I would rephrase it a bit (to make the statement clearer) and ...
- 1,375