**3**

votes

**2**answers

77 views

### Does the collection of coverings on a set $X$ form a lattice when ordered by refinement?

This is a follow-up question to this question, prompted by a comment in Todd Trimble's answer.
Let $X\neq \emptyset$ be a set. We say that $U\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a proper ...

**1**

vote

**1**answer

68 views

### Order on the collection of coverings

Let $X\neq \emptyset$ be a set. We say that $U\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a proper covering if
$\bigcup U = X$, and
for $a\neq b\in U$ we have $a\not\subseteq b$.
Let $\text{...

**3**

votes

**0**answers

51 views

### Matroid Representation of the Antichains of a Poset

Introduction
I am studying a problem in which the antichains of a poset are of key importance. They are naturally geometrically embedded as vectors in the space $\mathbb{R}^P$, where $P$ is the poset,...

**7**

votes

**1**answer

282 views

### Unique factorization of posets

Given two finite posets $P$ and $Q$, we can form the direct product poset $P \times Q$ whose elements are pairs $(p,q) \in P \times Q$ with $(p,q) \leq (p',q')$ if $p \leq p'$ and $q \leq q'$. Let us ...

**0**

votes

**0**answers

59 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?
...

**2**

votes

**1**answer

158 views

### Posets (partially ordered sets) in equational logic

I know about equational logic, cf. https://en.wikipedia.org/wiki/Lattice_(order)#Lattices_as_algebraic_structures, and understood that lattices are expressed equationally, i.e., in terms of equational ...

**4**

votes

**1**answer

157 views

### Is there a short expression for height and width of product and coproduct of posets?

I am trying to derive some basic relations for the height and width of the direct product and the coproduct of posets. I feel that these are very basic and should be written somewhere, however, I ...

**0**

votes

**0**answers

59 views

### Quick way to compute Ehrhart polynomial of Young diagram posets?

Using the hook formula, it is easy to compute the volume of order polytopes obtained from posets with partition shape, since this is the same as the number of linear extensions.
To my knowledge, ...

**9**

votes

**1**answer

172 views

### Are these two quotients of $\omega^\omega$ isomorphic?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say $f\simeq_{\text{fin}} g$ if there is $n\in \omega$ such that $f(k) = g(k)$ for all $k\geq n$.
...

**4**

votes

**0**answers

83 views

### Face structures of chain polytopes

For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain $p_1<\ldots<...

**3**

votes

**1**answer

154 views

### Can a length n distributive lattice be embedded into Bn?

Let $\mathcal{L}$ be a finite distributive lattice, then it is known that it can be embedded into a finite boolean lattice (see theorem 8.5. p91 in this note).
Let $n$ be the length of $\mathcal{...

**3**

votes

**0**answers

83 views

### Weighted maximal number of disjoint singly-generated ideals in the divisibility poset for $\{1,2,\ldots,n\}$

In the mathoverflow question here the asymptotic growth of antichains in the divisibility poset ${\cal P}_n$ of the set of natural numbers $\{1,\ldots,n\}$ is considered. I have a somewhat dual ...

**4**

votes

**2**answers

107 views

### What are bounds for the number of monotone functions $M:P\rightarrow T$ where $P$ is a finite poset and $T$ is a finite totally ordered set?

For the case where $P=\{0,1\}^n$ and $T=\{0,1\}$ the number of such functions is called the $n$-th Dedekind number and I discovered that there is large literature on determining bounds for these ...

**5**

votes

**0**answers

81 views

### Minimal Hausdorff topologies compatible with a bunch of functions

Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\...

**1**

vote

**0**answers

32 views

### Form of binary function over poset that is monotone over first and antitone over second argument

if I have a partially ordered set $P$, and I have a function $f: P \times P \to \mathbb{R}$ that is monotone over the first and antitone over the second argument, i.e. for any $a,b,c \in P$
$a ≤ b \...

**1**

vote

**0**answers

75 views

### reconstructing a linear order corrupted by noise

Suppose we have a partial order (efficiently computable), $\leq$, on $\mathbb{R}^n$, a set $S \subset \mathbb{R}^n$, and let $\rho$ be the standard Euclidean metric. We want to find a set $S^\prime = ...

**9**

votes

**0**answers

83 views

### Ideals in strong Bruhat order

Recall that a (lower) ideal in a poset $(P, \le)$ is a subset $I\subset P$ such that $x\in I \Rightarrow y\in I$ for all $y\le x$. In am interested in ideals in finite Coxeter groups $W$ equipped ...

**5**

votes

**1**answer

114 views

### An equivalent definition for the adiamond lattices

A lattice is called adiamond if it admits no sublattice equivalent to the diamond lattice $M_3$ below:
The top interval of a lattice is the interval between the meet of all the maximal elements and ...

**1**

vote

**1**answer

156 views

### Quotients of posets

Let $\mathbf{Poset}$ denote the category of partially ordered sets and order-preserving maps. Does $\mathbf{Poset}$ have quotients?

**5**

votes

**1**answer

140 views

### ACC (DCC) implies upper (lower) sets are upper (lower) closure of antichains?

I have read around (e.g. in Wikipedia) that if the ascending (descending) chain condition holds, all upper (lower) sets are the upper (lower) closure of an antichain, but I cannot find a proof.
More ...

**6**

votes

**2**answers

262 views

### Is this algebra isomorphic to an incidence algebra?

This question is motivated by trying to establish a converse to Theorem 7.8 of our paper.
I have a finite poset $P$ with the following properties:
$P$ has binary meets (and hence a least element).
$...

**1**

vote

**0**answers

89 views

### Properties of Coefficients of Order Polynomials [closed]

I am working on a problem involving determining the order polynomial $\Omega_P(k)$ of a partial order $P$, which counts the number of order-preserving transformations/maps from $P$ to the $k$-chain (...

**2**

votes

**1**answer

110 views

### Surjectively rigid partially ordered sets

This question is related to a question recently asked by Joel David Hamkins.
Let $(P,\leq)$ be a poset. We call it surjectively rigid if the only order-preserving surjective map $f:P\to P$ is the ...

**7**

votes

**0**answers

237 views

### Pattern Avoidance in Poset Permutations

I am not sure if it is appropriate to use MathOverflow to publicize a conjecture, but I think this is an interesting question and I have no real ideas of how to solve it.
A permutation on a $n$-...

**0**

votes

**0**answers

67 views

### An equality for the trace of the join of a non-degenerate indecomposable system of projections in a finite factor

Let $M \subset B(H)$ be a finite factor (see for example here p2, or there) with a trace $tr$.
The subset of projections of $M$ is naturally a lattice, noted $(\mathcal{P}(M), \wedge, \vee)$.
A ...

**3**

votes

**1**answer

185 views

### Is the top interval of a finite distributive lattice, a boolean lattice?

Let $(L,\wedge,\vee)$ be a finite distributive lattice, and let $1$ its greatest element.
An element $a \in L$ is called maximal if $a \le a' < 1$ implies $a = a'$.
Let $b$ be the meet of all the ...

**11**

votes

**1**answer

259 views

### Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property

What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis.
$\mathbb{P}$ preserves stationary subsets ...

**8**

votes

**2**answers

300 views

### Posets isomorphic to their endomorphism poset

Let $(P,\leq)$ be a poset. We set $$\text{End}(P)=\{f: P\to P: f\text{ is order-preserving}\}$$ and order $\text{End}(P)$ pointwise.
Is there a poset with more than 1 point such that $P\cong \text{...

**8**

votes

**1**answer

147 views

### Optimal strategy for game of 'online sorting' into a poset

Consider a single-player game played with an arbitrary finite poset, and a random number generator with a known distribution:
Each turn, the RNG produces a number, and the player must assign that ...

**7**

votes

**1**answer

200 views

### Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.
Question: Suppose $\mathbb{P}$ is a separative partial order which is $\...

**9**

votes

**0**answers

121 views

### A family of posets

Consider the family of all (finite) posets that can be obtained by repeatedly applying one of the following three operations (starting e.g. with the empty poset):
(O1) Disjoint union of one or more ...

**0**

votes

**1**answer

84 views

### Dedekind-MacNeille completion of the strictly increasing members of $\omega^\omega$

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ ordered by $f\leq g$ iff $f(n) \leq g(n)$ for all $n\in \omega$.
Set $K = \{f\in \omega^\omega: m<n\in \omega \implies f(m)&...

**3**

votes

**1**answer

177 views

### Order dimension of $\omega^\omega/(fin)$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we say $f\simeq g$ if and only if $\exists N \in \omega$ such that $f(n) = g(n)$ for all $n\geq N$.
...

**0**

votes

**1**answer

98 views

### Priestley topologizability and connected components

This question is in the spirit of another older question.
We say that a poset $(P,\leq)$ is Priestley-topologizable if there is a topology $\tau$ on $P$ such that $(P,\leq,\tau)$ is a Priestley space....

**8**

votes

**1**answer

173 views

### Extending subsets to supersets in different ways

We are given a collection of sets $A_1,\ldots,A_s$, pairwise different and each of cardinality $k$, and a collection of sets $B_1,\ldots,B_s$, pairwise different and each of cardinality $l>k+1$, ...

**1**

vote

**2**answers

131 views

### Poset-enrichment of abelian categories

Let $\mathsf{A}$ be an Abelian category (perhaps vector spaces or modules over your favorite ring), and let $\mathsf{A}(x,y)$ denote the set of morphisms in $\mathsf{A}$ from an object $x$ to another ...

**1**

vote

**1**answer

101 views

### Interval topology and order convergence topology

Throughout this post, let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\...

**0**

votes

**2**answers

116 views

### Continuous image relation on topological spaces

Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by
for $X, Y \in \text{...

**2**

votes

**3**answers

227 views

### Antichain on $\mathcal{P}(\omega)/fin$ of cardinality $2^{\aleph_0}$?

We define an equivalence relation on $\mathcal{P}(\omega)$: for $x,y\in\mathcal{P}(\omega)$ we say $$x\simeq_{fin} y \text{ iff there is } n \in \omega \text{ such that }
x\setminus \{0,\ldots,n\} = y ...

**3**

votes

**1**answer

87 views

### Is $\{0,1\}^\omega$ the order-preserving image of $\{0,1\}^\omega$ modulo some finiteness relation?

Consider the following equivalence relation on $\{0,1\}^\omega$:
$x\simeq y$ iff there is $n\in\omega$ such that $x(k)=y(k)$ for all $k\in\omega$ with $k\geq n$.
It is easy to see that ...

**1**

vote

**2**answers

114 views

### Completion of a single totally ordered down-set

This is a follow-up question to Complete sets of incompatible totally ordered down-set in a partially ordered set.
Let $(P,\leq)$ be a partially ordered set such that for every $p\in P$ the set $\{q\...

**2**

votes

**2**answers

209 views

### Complete sets of incompatible totally ordered down-set in a partially ordered set

Let $(P,\leq)$ be a partially ordered set. A down-set is a set $d\subseteq P$ such that $x\in d$ and $x'\in P, x'\leq x$ imply $x'\in d$. If the down-set is totally ordered, we say it is a totally ...

**10**

votes

**3**answers

449 views

### Is a distributive lattice planar iff it admits no B3 sublattice?

A finite lattice is planar if it admits a Hasse diagram which is a planar graph (we want the Hasse diagram to be represented in the plane so that the $y$-coordinate of each element respects the order)...

**3**

votes

**1**answer

147 views

### Directed Hypercube Minimal Cuts

If $[n]:=\{1,2,\ldots, n\}$ for some $n\in\mathbb{N}$, then the hypercube digraph of dimension $n$, denoted $H_n$, is the graph whose set of vertices is the power-set $\wp([n])$ where two vertices $U,...

**2**

votes

**0**answers

44 views

### Decidability of first order theory of subclasses of posets

Is the first order theory of finite posets known to be undecidable?
Does anyone know a survey about such results?

**1**

vote

**1**answer

91 views

### Order-preserving images of $(\mathcal{P}(\kappa),\subseteq)$

Is there a cardinal $\kappa \neq \emptyset$ and a connected poset $P$ of cardinality $\leq \kappa$ such that there is no surjective order-preserving map from $(\mathcal{P}(\kappa),\subseteq)$ onto $P$?...

**4**

votes

**2**answers

143 views

### Image of poset with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology $\tau_{\text{int}}(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\...

**2**

votes

**1**answer

140 views

### Product of posets with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\...

**5**

votes

**1**answer

213 views

### Is there a standard name for this poset

I've run into the following poset and I would expect it has a standard name. Let $n\geq k\geq 0$. Then $P_{n,k}$ consists of all $k$-element subsets of $\{1,\ldots,n\}$ ordered by $X\leq Y$ if $X=\{...

**2**

votes

**1**answer

140 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 ...