# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### Quotients of posets

Let $\mathbf{Poset}$ denote the category of partially ordered sets and order-preserving maps. Does $\mathbf{Poset}$ have quotients?
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### 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 ...
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### 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 ...
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### 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 $n\geq N$. ...
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### 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....
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### 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 $l>k+1$, ...
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### 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 ...
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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 ... 1answer 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 ... 2answers 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\... 2answers 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 ... 3answers 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)... 1answer 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,... 0answers 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? 1answer 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?... 2answers 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\},$$where \downarrow x = \{y\... 1answer 140 views ### Product of posets with Hausdorff interval topology Given a poset (P,\leq) the interval topology on P is generated by$$\{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 $\... 1answer 213 views ### Is there a standard name for this poset I've run into the following poset and I would expect it has a standard name. Let$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=\{...
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 ...