# 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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### Are there overwhelmingly more finite posets than finite groups? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
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### Listing all posets on 9 points?

I'm looking for a list of all (non-isomorphic) posets on 9 points. I know there are 183231 of them (OEIS A000112), but in order to progress with a problem I'm working on, I'd need the posets ...
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### ${\frak b}$ and ${\frak d}$ in the Rudin-Keisler preordering

If $(Q,\leq)$ is any preordered set (that is, $\leq$ is a reflexive and transitive, but not necessarily anti-symmetric relation), then we say that $S\subseteq Q$ is unbounded if for all $q\in Q$ ...
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### Minimizing the set of monochromatic edges

For sets $A, B$ we write $B^A$ for the set of all functions $f:A\to B$. Let $H = (V,E)$ be a hypergraph such that $V,E\neq\varnothing$ and $|e| \geq 2$ for all $e\in E$. Let $\kappa>1$ be a ...
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### Sum of order polynomials of a set of posets

Let $n\in \mathbb{Z}_{>0}$. For every subset $S\subseteq \left[ n-1\right]$ we define a poset $P_S=\left([n],\le_{P_S}\right)$ given by the covering relation $\lessdot$ which is defined as \begin{...
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### Can one characterize maximal antichains in terms of distributive lattices?

This is inspired by the recent question Verification of a maximal antichain The celebrated duality between finite posets and finite distributive lattices has several nice formulations. One of them ...
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### Verification of a maximal antichain

In order theory, an antichain (Sperner family/clutter) is a subset of a partially-ordered set, with the property that no two elements are comparable with each other. A maximal antichain is the ...
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### Maximal order of an order-preserving map

Let $X$ be a finite partially ordered set, let $f\colon X\to X$ be an order-preserving map [edit: meaning $x\le y\implies f(x)\le f(y)$], and let $x_0$ be an initial point. Define $x_n = f(x_{n-1})$ ...
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### Expected height of a poset?

I am interested in any known results/empirical studies done on the average height of a poset with $N$ elements. Obviously this would depend on how that poset relation was randomly defined, however, at ...
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### On the width of the Catalan monoid and the rank of K-groups of the Furstenberg transformation group

The semigroup algebra of the Catalan monoid is isomorphic to the incidence algebra of $P_n$, where $P_n$ is the poset consisting of subsets of { 1,...,n } where for two subsets $X \leq Y$ if and only ...
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### A meet-semilattice with top element that is not a lattice?

I am reading Francis Borceux’s “Handbook of Categorical Algebra I” and on page 135 it says In particular a finite version of 4.2.5 does not hold: a finitely complete and well-powered category ...
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### Periods of Coxeter transformation associated to root posets

$\DeclareMathOperator\Co{Co}$Let $P$ be the root poset associated to a simple Lie algebra. Let $L=L(P)$ denote the distributive lattice of order ideals of $P$ and let $\Co_L$ denote the Coxeter matrix ...
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### When are quotients of the Boolean lattice dissective?

Let $B_n$ be the Boolean lattice and $G$ a subgroup of $S_n$ acting on $B_n$. Let $P_G=B_n/G$ denote the quotient poset. Question 1: When is $P_G$ a lattice? When is it distributive in that case? An ...
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### Classification of posets that are quotient posets of the Boolean lattice

Quotient posets of the Boolean lattice $B_n$ have interesting properties and are for example discussed in chapter 5 of Stanley's book on algebraic combinatorics. $B_n/G$ for a subgroup $G$ of the ...
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### Shellability and order filters in the partition lattice

Choose $n\in\mathbb N$. Let $B$ be a non-empty subset of $[n]:=\{1,2,\dots,n\}$. Consider the set of partitions of the set $[n]$ with exactly $|B|$ parts such that each part has exactly one member ...
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### Non-commutative version of the order dimension of a poset

I view the order dimension of a poset $P$ as an inherently commutative notion. On the one hand, it can be defined via realizers, which I find fairly intuitive from an order-theoretic viewpoint. On the ...
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### The average size of downward closed family of the subsets of $[n]$ is at most $n/2$?

I learned that the average size in any ideal of subsets of $[n]$ is at most $n/2$, but I think the downward closed family of the subsets of $[n]$ also satisfied. I want to know how to proof it or it ...
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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. ...
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 ...
### 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 ...