# Questions tagged [posets]

A poset or partially ordered set is a set endowed with a partial order, meaning a binary relation $\leq$ which is reflexive ($x \leq x$ for all $x$), antisymmetric ($x\leq y$ and $y\leq x$ implies $x=y$), and transitive ($x\leq y$ and $y\leq z$ implies $x \leq z$).

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### Posets which extend centered sets to filters

(Post cross-posted from math.se.) Suppose $(\mathcal O, \leq)$ is an arbitrary poset. Let us say that $\mathcal O$ is compact if every $\mathcal C\subseteq\mathcal O$ which is centered (any finite ...
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### Do you recognise this setup of structure on a poset?

The setup is that we have a finite poset $P$, with a multiplicative rank function $r_{xy}:P\times P\rightarrow \mathbb{N}$, and a symmetric pairing $\langle\ ,\ \rangle:P\times P\rightarrow\mathbb{N}$....
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### Poset-troids …?

In many respects, spanning tree : graph :: linear extension : poset For instance, the number of spanning trees/linear extensions is a measure of the "richness" or "complexity" of the graph/poset. ...
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### Number of minimal elements of product order

Consider three sets: $A = \{1,2,\dotsc,n_A\}$, $B=\{1,2,\dotsc,n_B\}$, and $C=\{1,2,\dotsc,n_C\}$, where $n_A, n_B, n_C \ge 2$. Define a product order (which is a partial order) on the cartesian ...
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### Is this ordering on the set of all covers of $\omega$ a (complete) lattice?

Let ${\frak C} \subseteq {\cal P}({\cal P}(\omega))$ be the collection of all covers of $\omega$ (that is, ${\cal C} \in {\frak C}$ iff $\bigcup {\cal C} = \omega$.) We define the following binary ...
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In my recent work I stumbled across a problem of this type: G with two partial oders $\leq$ and $\preceq$ on every set, i.e. for every $n \in \mathbb{N}$ $A_n \subset A_{n+1}$ and $(A_n, \leq)$ and $... 0answers 119 views ### Is the order complex of open Bruhat intervals polytopal? Let$P$be the Bruhat order of a Coxeter group, and let$s<t$in$P$. The set$\Delta(s,t)$of all chains of the open interval$(s,t)$(called the order complex of$(s,t)$) is a simplicial complex. ... 1answer 139 views ### Bounding and domination numbers for relation$\leq$modulo$\omega$-nullsets We say that$A\subseteq \omega$is a nullset if $$\lim\sup_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$ Let$\omega^\omega$denote the set of functions$f:\omega\to\omega$. We define a pre-ordering ... 1answer 70 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$\...
A Kreweras walk of length $3n$ is a word consisting of $n$ $A$'s, $n$ $B$'s, and $n$ $C$'s such that in any prefix there are at least as many $A$'s as $B$'s, and at least as many $A$'s as $C$'s. For ...