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 $x$) and transitive ($x\leq y$ and $y\leq z$ implies $x \leq z$).

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Is the order convergence topology on a poset always Hausdorff?

In this post two topologies on a poset $(P,\leq)$ were defined: the interval topology $\tau_i(P)$ and the order convergence topology $\tau_o(P)$. It turns out that $\tau_i(P)$ is always $T_1$ and ...
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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 ...
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Characterizing Posets by Functions Into Natural Numbers

Let $P$ be a poset and denote by $Hom(P, \mathbb N)$ the set of all monotone functions from $P$ to natural numbers $\mathbb N$. Under what conditions on $P$ Is it possible to recover the order on $P$ ...
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529 views

Order homomorphism functions on $\omega_1$

Let $\omega_1$ be the first uncountable ordinal, same as the set of all countable ordinals. Let $F$ be the set of all functions $f$ from $\omega_1$ minus singleton $0$ into $\omega_1$ that are (a) ...
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129 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 ...
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620 views

Bruhat order and the Robinson-Schensted correspondence

The Robinson-Schensted correspondence is a bijection between elements of the symmetric group $S_n$ and pairs of standard tableaux of the same shape. The symmetric group is partially ordered by the ...
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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 ...
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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 ...
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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 ...
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How exactly does Schützenberger promotion relate to Striker-Williams promotion?

Schützenberger promotion, studied (for example) in Richard Stanley, Promotion and Evacuation, 2009, is a permutation of the set of all linear extensions of a finite poset. Since one can identify the ...
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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 ...
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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 ...
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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 ...
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380 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 ...
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42 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?
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75 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 ...
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1answer
102 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 ...
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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 = ...
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200 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 ...
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Universal and left-factoring order-preserving maps

Trying to get a different angle for the question Fixed points and universal maps for posets, I want to compare universal maps to a different kind of functions. First recall that for posets $P,Q$ an ...
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Is the homomorphism poset directed if the codomain is directed?

Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. (We call a function $f:P\to Q$ order preserving if $x\leq y$ in $P$ ...
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212 views

When is the homomorphism poset between posets a lattice?

Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. There is a natural ordering relation on $\text{Hom}(P,Q)$ given by $f\leq ...
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177 views

Fastest algorithm to compute the width of a poset

An colleague recently came to me with a problem concerning the scheduling of tasks in the presence of constraints (of the kind: task $x$ can't begin until task $y$ has been completed). It turned out ...
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118 views

Does the category of posets have pushouts and pullbacks? [duplicate]

Let $\mathbf{Poset}$ be the category of partially ordered sets with order-preserving maps. Does $\mathbf{Poset}$ have both pushouts and pullbacks?
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Question on Posets and open sets [closed]

i'm sorry if my question is really trivial but this one is really bugging me out.. So let's have a partially ordered set $I$ with the topology in which the open sets are the increasing ones: $i\in U$ ...
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Computing basis of a lower set given basis of complementary upper set

In a poset $P$, $U\subseteq P$ is an upper set when for all $x\in U$, we have $y\ge x$ implies $y\in U$. Any subset of $P$ generates an upper set, and the basis of an upper set $U$ is the smallest ...
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286 views

Classification of countable posets?

Is there a classification of countable posets where between each two comparable elements there is a third element between them?
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871 views

Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorem. Suppose $P$ and $Q$ are posets ...
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Fixed points and universal maps for posets

In a recent post about f.p.p. for poset products, @M.Mirabi brought back an old-standing problem about the fixed point property of the product of two arbitrary posets which already enjoy the fixed ...
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Directed subposet of a poset containing the minimal elements

The following appears naturally in a certain context: Let $P$ be a graded partially ordered set. Let $M$ be the subset of minimal elements of $P$. Define subsets $E_i$ inductively as follows: First, ...
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a dcpo seen as a category: when does a dcpo map induce a functor with an adjoint?

Take two posets $A, B$ (partially ordered sets). Now consider these posets to be categories $Cat(A), Cat(B)$ respectively. Consider a map from $A$ to $B$, $f: A \rightarrow B$. This can be seen as ...
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The category of categories and adjunctions

What is known about the category that has small categories as objects and adjunctions as morphisms? Obviously, it has neither terminal nor initial objects. But what about other kinds of limits? Are ...
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178 views

Principal Order Ideals in the Weak Bruhat Order

Let $\sigma\in S_n$ be a permutation on $n$ elements, and $\mathrm{Inv}(\sigma):=\{(i,j) : 1\leq i<j\leq n\text{ and }\sigma(i)>\sigma(j)\}$ be its set of inversions. In the weak order on ...
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94 views

Krull dimension of dense extensions

Let $A$ be a boolean algebra and let $B\leq A$ be a boolean sub-algebra which is dense (for all $0\neq a\in A$, there is a $0\neq b\in B$ such that $b\leq a$). We suppose also that $B$, as a partially ...
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128 views

Galois Connections: algorithmic generation

Given two finite posets $P,Q$, is it known any algorithm to count and/or generate every Galois Connection between $P$ and $Q$ ? I'm looking for references about this problem.
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178 views

Maximal number of antichains of a connected poset

Assume we have a connected poset $P$ of $n$ elements, I am searching to know what is the maximal number of antichains such a poset can have? $2^n$ is obviously an upper bound, and my feeling is that ...
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Strength of claims about extensions of partial preorders and orders to linear ones

Consider these two axioms: Every partial order extends to a linear order. Every partial preorder (reflexive and transitive relation) extends to a linear preorder while preserving strict orderings: ...
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Matching in the Boolean Algebra

We consider the Boolean Algebra of the set $[n]=\{1,\dots,n\}$ and the matching $\psi$ from the $m+1$ subsets of $[n]$ into the $m$ subsets of $[n]$ for $(n+1)/2\leq m+1\leq n$, which is used to show ...
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Subposets of small Dushnik-Miller dimension

The Dushnik–Miller dimension of a partial order $(P,{\leq})$ is the smallest possible size $d$ for a family ${\leq_1},\ldots,{\leq_d}$ of total orderings of $P$ whose intersection is ${\leq}$, ...
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444 views

Minimal (semi)lattice containing a given poset

For a given poset, (I think that) it is easy to construct the minimal join-semilattice containing that poset. I wonder whether the minimal lattice containing that poset is also easy to construct. I ...
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references for properties/examples of breadth in (semi)lattices

This is in some sense following up on my earlier question and the answer given by NN. I am currently revising the paper which used the condition mentioned in my question. It was pointed out in NN's ...
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2answers
114 views

Representation of Banach spaces partially ordered by solid, normal, minihedral cones

I've been using the representation result below, from Krasnosel'skij/Lifshits/Sobolev; Positive Linear Systems---The Method of Positive Linear Operators. Heldermann Verlag, 1989. Theorem. Let $E$ be ...
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About subposet of Levy collapse

Let $\lambda>\kappa$ and $\operatorname{Coll}(\kappa, \lambda)$ be the poset collapsing $\lambda$ to $\kappa$. Pick a subposet $P$ which is $\lt\kappa$-closed and of size $\lambda$. Can we say that ...
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Suprema and infima in spaces ordered by non-normal cones

Background We shall call a subset $V_+ \subseteq V$ of a Banach space $V$ a cone if $V_+$ is closed, $\alpha V_+ \subseteq V_+$ for all $\alpha \geqslant 0$, and $V_+ \cap (-V_+) = \{0\}$. Cones ...
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Is the Poset of Graphs Automorphism-free?

For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$. ...
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Rank generating functions on a graded poset and a linear extension of it

Let $A$ be a possibly infinite ensemble and let $\leq$ be a partial order between elements of $A$. We denote $P_{\leq}=(A,\leq)$ the corresponding poset. Furthermore, we suppose that $P$ is graded, ...
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282 views

Count of lattices on finite set

Let $p(n)$ denote count of lattices on finite set $G$, $|G|=n$ (without isomorphism). It's know closed formula for $p(n)$? It's clear, that $1 \leq p(n)$ and also that $p(n-1) \leq p(n)$ for $n \geq ...
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90 views

Generalized connected components decomposition for Priestley spaces

Preliminaries A partially ordered space is both a poset and a topological space. It has connected components both as a topological space, and connected components as a poset, i.e. the maximal ...
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144 views

Linear order extensions on (nonabelian) groups

If $G$ is a group with a (left) linear order, does every (left) partial order on $G$ extend to a (left) linear order? The answer is affirmative on abelian groups, where being torsion-free is ...
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Are the connected components of a Priestley space closed?

Preliminaries A Priestley space is both a poset and a topological space. The topologically connected components of the space are trivially closed. (They are just the points of the underlying set.) But ...