All Questions
8 questions
9
votes
1
answer
542
views
Reference request: number of antichains of a partially ordered set
Let $\mathbb{N}$ denote the set of all positive integers. For each $n \in \mathbb{N}$, define the set $$ P_n = \{ (a,b) \in \mathbb{N} \times \mathbb{N} : 1 \leq a \leq b \leq n \} $$ and consider the ...
- 563
5
votes
0
answers
201
views
Is this "trimming" of a supersolvable semimodular lattice known?
Let $L$ be a finite (upper) semimodular lattice. Recall that this means $L$ is graded and its rank function $\rho\colon L \to \mathbb{N}$ satisfies
$$ \rho(x) + \rho(y) \geq \rho(x\vee y)+\rho(x \...
- 24.2k
2
votes
1
answer
94
views
Request for literature recommendations on isotonic mappings
An isotonic mapping is a function between two partially ordered sets that preserves the ordering between the elements. Specifically, given two partially ordered sets $(X,\le)$ and $(Y,\le)$, a ...
- 21
7
votes
0
answers
118
views
Dimension of a union of downsets
We have established the following result regarding the Dushnik–Miller dimension of posets.
Let $P$ be a poset with downsets $C, D \subseteq P$. If the dimensions of $C$ and $D$ are $m$ and $n$, ...
- 316
0
votes
0
answers
131
views
terminology: monotone maps of posets such that the image of a lower set is a lower set
How are called in combinatorics
monotone maps of partially ordered sets such that the image of a lower set is a lower set, i.e. closed (or open) maps of finite topologies? Is there a classification ...
- 113
1
vote
0
answers
731
views
"Downward closed" relation on a poset
I say that a relation $R$ on a poset $P$ is downward closed if for each $(x,y)\in R$, and $x'\le x$, then $(x',y)\in R$.
Is there a reference where this thing is studied, maybe under a different name?
...
- 13.6k
3
votes
1
answer
211
views
Terminology question for maps between posets
Let $P$ and $Q$ be two poset (partially ordered sets) and $\phi : P \to Q$ an order-preserving function.
I would like to know whether there is a name and perhaps a different characterizations of such ...
- 408
5
votes
1
answer
178
views
Reference for statement that almost every $n$-element partial order has trivial automorphism group
I'm looking for a reference for the statement that almost every partial order on $n$ elements has trivial automorphism group. I've been told that this is a folklore result. Does anyone know of a ...
- 462