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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 $x \leq z$).

2
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2answers
119 views

Infima and suprema in the “transfer” function ordering

Let $X,Y$ be sets, $f, g:X\to Y$ be functions. We say $u:Y\to Y$ is a transfer function for $g$ to $f$ if $$f = u \circ g.$$ In that case we write $f \leq_t g$. Let $\mathrm{Fct}(X,Y)$ denote the ...
2
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0answers
45 views

Obtaining the reduced incidence algebra in QPA

Given a finite poset $P$ (we can assume it is connected), the reduced incidence algebra of $P$ is the subalgebra of the incidence algebra of $P$ consisting of functions constant on isomorphic ...
16
votes
1answer
510 views

How is this fixed point theorem related to the axiom of choice?

I'm hoping the answer to this is well-known. Let $X$ be an ordered set (i.e. poset). An inflationary operator $f$ on $X$ is a function $f: X \to X$, not necessarily order-preserving, such that $f(x) ...
6
votes
1answer
224 views

Does there exist a full and faithful embedding of $\mathsf{Poset}$ in $\mathsf{Set}$?

Does there exist a full and faithful embedding of the category of posets into the category of sets? I suspect no, but I don't know how to prove or disprove this.
0
votes
3answers
189 views

Is every graph an incomparability graph?

Let $G=(V,E)$ be a simple, undirected graph. Is there a partial ordering $\leq\subseteq (V\times V)$ with the following property? $$\{v,w\} \in E \text{ if and only if } v||y$$ (We write $v||w$ in ...
11
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1answer
154 views

Connected incomparability graph

Let $X$ be a finite set equipped with a partial order. (Not a preorder: $a < b$ implies $b \not< a$.) The corresponding incomparability graph has vertex set $X$ with an edge between two points ...
11
votes
1answer
244 views

“Scott completion” of dcpo

If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup_I a_i \in U $ for ...
8
votes
2answers
274 views

Surjective order-preserving map $f:{\cal P}(X)\to \text{Part}(X)$

Let $X$ be a set, and let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ ...
4
votes
1answer
156 views

Embedding ordinals with the order topology into connected $T_2$-spaces

Is there a limit ordinal $\kappa_0$ with $\kappa_0 \lt 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\lt 2^{\aleph_0}$ there is a connected $T_2$-space $X_\...
3
votes
0answers
151 views

The name for injective map $f:\mathbb{N}\rightarrow\mathbb{N}$ with $f(n)\geq n$ property

What is the name for map $f:\mathbb{N}\rightarrow\mathbb{N}$ (from natural numbers into natural numbers) with the following propeties: 1) $f$ is injective 2) $f(n)\geq n$ for every $n$?
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votes
1answer
114 views

Upward generators of $[\omega]^\omega$

If $(P,\leq)$ is a poset and $S\subseteq P$ we let $$\uparrow S = \{p\in P: p\geq s\text{ for some }s\in S\}.$$ Let $([\omega]^\omega,\subseteq)$ denote the collection of infinite subsets of $\omega$,...
7
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0answers
190 views

Counting hyperplane arrangements up to combinatorial equivalence, simple examples and history

Two arrangements of (affine) hyperplanes in $d$-dimensional Euclidean space are combinatorially isomorphic (or combinatorially equivalent) if they have isomorphic posets of faces. Counting the ...
4
votes
1answer
127 views

Order convergence vs topological convergence in partially ordered sets

Short version of the question. If $(P,\leq)$ is a partially ordered set (poset), a topology denoted by $\tau_o(P)$ can be defined (see below). There is also another notion of convergence, called order-...
3
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0answers
66 views

Embedding finite partially ordered sets into antisymmetric monoids

I am wondering if there is an easy answer to the following question: Let us consider a finite partially ordered set $P$. It is clear that there exists a $k$ such that there is an order embedding $P\...
3
votes
0answers
52 views

(When) is the Dedekind-MacNeille completion of a po-set Hausdorff?

Let $X$ be a p.o. Consider the topology on $X$ generated by $$U_{x}^{-}:=X\setminus (x\uparrow),\quad U_{x}^{+}:=X\setminus (x\downarrow), \quad x\in X$$ Throughout this discussion I shall refer to ...
0
votes
1answer
103 views

Ordered group acting freely on partially ordered set

Let $(G, <)$ be a totally ordered group, and let $<$ be left-invariant. Let $G$ act (freely?) on a partially ordered set $(S, <)$, such that this group action preserves the ordering: $$ s_1 &...
8
votes
1answer
197 views

Is the measurable space $(\omega_1,\mathcal{P}(\omega_1))$ separable?

Here $\omega_1$ is the first uncountable ordinal, and $\mathcal{P}(\omega_1)$ denotes the power set of $\omega_1$. Separable means countably generated as a $\sigma$-algebra.
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0answers
40 views

A weaker locally convex topology on a pospace

A pospace is a topological space $X$ endowed with a closed partial order $\le$. A pospace $X$ is locally convex if it has a base of the topology consisting of open order-convex sets. A subset $A$ of a ...
0
votes
1answer
171 views

$\text{Max}\big(\text{Sub}(\text{Sym}(\omega))\setminus \{\text{Sym}(\omega)\}\big)$

If $G$ is any group, then by $\text{Sub}(G)$ we denote the collection of all subgroups, ordered by $\subseteq$. If $(P,\leq)$ is a partially ordered set we let $\text{Max}(P)$ and the set of maximal ...
2
votes
1answer
94 views

Boolean completion of a partially ordered set

Given a poset $(P, \leq)$, is there a complete Boolean lattice $B$ and an order-preserving map $i_P: P\to B$ such that for any complete Boolean lattice $B'$ and order-preserving map $f: P\to B'$ ...
6
votes
2answers
236 views

Is an Eulerian lattice shellable?

The notion of Eulerian lattice generalizes the notion of face lattice of a convex polytope. (Bruggesse-Mani): The boundary complex of a convex polytope is shellable. (Björner-Wachs): A poset is ...
4
votes
2answers
92 views

Order-embedding, but no lattice embedding between distributive lattices

Let $L$ be the power set lattice ${\cal P}(\{0,1,2\})$. It is clear that there is an order-preserving injective map from $M_3$ into $L$, but no injective lattice homomorphism (because $L$ is ...
9
votes
3answers
352 views

Does the lattice of all topologies embed into the lattice of $T_1$-topologies?

Let $\kappa$ be an infinite cardinal, and let $\text{Top}(\kappa)$ be the lattice of all topologies on $\kappa$, ordered by $\subseteq$. Let $\text{Top}^{T_1}(\kappa)$ be the lattice of all $T_1$-...
7
votes
0answers
95 views

Dimension of a union of downsets

We have established the following result regarding the Dushnik–Miller dimension of posets. Let $P$ be a poset with downsets $C, D \subseteq P$. If the dimensions of $C$ and $D$ are $m$ and $n$, ...
7
votes
1answer
326 views

Structure of $Hom(L_1,L_2)$, where $L_i$ are distributive lattices

Is there known structures/ or has there been studies on $Hom(L_1,L_2)$ of distributive lattices? Could it be made into a lattice naturally? Is there any structure on the set of ring valued functions $...
3
votes
1answer
133 views

Maximal elements in the partially ordered set of image spaces

If $(X,\tau)$ is a topological space, let $\text{Im}(X)$ denote the collection of subsets $S$ of $X$ such that there is a continuous function $f:X\to X$ with $\text{im}(f) = S$. Is there a space $(X,\...
10
votes
0answers
139 views

Poset of nonvanishing minors of a matrix

This question was posed on MSE here three days ago, but hasn't gotten any answers or suggestions. I hope it's okay to ask it on MO, but if I should wait a little longer, please just let me know. Say $...
1
vote
0answers
49 views

What is known about the following partial order (subquotients of linearly ordered set)

Fix two numbers $n,k\in \mathbb{N}$ and abbreviate $[n]:= \{0,1,\dots,n\}$ (as linearly ordered set). Consider diagrams $(I,\varphi)$ of the form $$[n]\supset I\xrightarrow{\varphi}[k]$$ where $\...
6
votes
4answers
372 views

Strict and non-strict orderings

Consider a set $A$ equipped with two binary relations $\le$ and $<$, related in the appropriate ways for the strict and non-strict version of an ordering. One might make different choices about ...
3
votes
1answer
129 views

The Wallman and interval topologies on non-principal ultrafilters with the Rudin-Keisler preorder

If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow_{\leq} x) = \{p\in P: p\leq ...
3
votes
3answers
255 views

Countably infinite posets isomorphic to its intervals

Let $(P,\leq)$ be a countably infinite poset with the property that whenever $a<b\in P$ then $P\cong [a,b]$. Question. If $P$ does contain elements $a,b$ with $a<b$, does this imply that $P \...
0
votes
2answers
140 views

Is ${\cal P}(\omega)/\mathrm{(fin)}$ order-isomorphic to its intervals?

Let $a, b \in {\cal P}(\omega)/\mathrm{(fin)}$ with $a<b$. Do we have ${\cal P}(\omega)/(fin)\cong [a,b]$?
3
votes
1answer
162 views

Order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$

Is there an order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$? Or from ${\cal P}(\omega)/(fin)$ onto $[0,1]\cap \mathbb{Q}$?
4
votes
1answer
106 views

“Gaps” in the Rudin-Keisler ordering

If $(P,\leq)$ is a poset and $p\in P$, then we say that $p$ is the lower part of a gap there is $q \in P$, $q>p$ such that $[p,q] = \{p,q\}$. (This is equivalent to the statement that $(\uparrow p) ...
11
votes
2answers
478 views

What is known about ideal and divisibility lattices of GCD domains and their generalizations?

The divisibility relation "$a$ divides $b$", or concisely, $a \vert b$ defined over a commutative integral domain $R$ with identity induces a partial order on the multiplicative semigroup $R/R^{\times}...
2
votes
1answer
89 views

Is $({\cal P}(\omega), \leq_{\text{inj}})$ a distributive lattice?

For $A, B \subseteq \omega$ we set $A \leq_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a down-set of $B$. It is easy to see that $\leq_{...
5
votes
1answer
263 views

A different ordering on ${\cal P}(\omega)$

For $A, B \subseteq \omega$ we set $A \leq_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a down-set of $B$. It is easy to see that $\leq_{...
3
votes
1answer
140 views

Cardinality of maximal chains in the poset of ultrafilters with Rudin-Keisler ordering

Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by $${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:...
3
votes
1answer
112 views

Unboundedness number and domination number of a poset $(P,\leq)$

Suppose $(P,\leq)$ is a poset without maximal elements. For $X\subseteq P$ we set $X^u = \{p\in P: p \geq x \text{ for all } x\in X\}$ and call this the set of upper bounds of $X$. We say that $B\...
5
votes
0answers
89 views

Probability of a maximal chain in a random subposet of a finite poset

Let $P$ be a finite poset, and let $0<p<1$. Choose a random subposet $Q$ of $P$ by letting each $t\in P$ belong to $Q$ with probability $p$. What is the best way to compute the probability that $...
4
votes
1answer
93 views

Posets as graphs with the direct neighbor relation

Given any poset $(P,\leq)$ we define the "direct neighbor graph" as follows. Let $$E_P = \big\{\{a,b\}: (a<b \text{ or } a>b) \text{ and } \; ]\min\{a,b\},\max\{a,b\}[ = \emptyset\big\}.$$ It is ...
4
votes
0answers
210 views

Can infinite bounded distibutive lattices be “arbitrarily wide”?

I was always thinking, in an informal way, that the powerset lattices ${\cal P}(X)$ (where $X$ is an infinite set) are the "widest" bounded distributive lattices with respect to their height. (In ${\...
6
votes
0answers
146 views

Katetov ordering on ideals on $\omega$

Recall that a nonempty set ${\cal I}\subseteq {\cal P}(\omega)$ is a (set) ideal if $B\in{\cal I}$ and $A\subseteq B$ imply $A\in{\cal I}$, and $A,B \in {\cal I}$ implies $A\cup B\in {\cal I}$. By $\...
3
votes
1answer
225 views

Antichains of maximum cardinality: posets vs lattices

The following construction gives a poset such that no antichain has maximum cardinality: For $n\in\mathbb{N}\setminus\{0\}$, let "layer" $n$ consist of an antichain of $n$ points, and as for the ...
-1
votes
1answer
108 views

Covering property of complete distributive lattices

Let $(L,\land,\lor)$ be a complete distributive lattice. Given $x\neq y \in L$, is there a finite set ${\cal I}$ of closed intervals in $L$ such that no member of ${\cal I}$ contains both $x$ and $y$,...
2
votes
1answer
138 views

Adjoints of the interval topology functor

Given a poset $(P,\leq)$ 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 $\downarrow x = \{y\in P: y\...
2
votes
1answer
76 views

Hausdorff interval topology on distributive lattices

Given a poset $(P,\leq)$ 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 $\downarrow x = \{y\in P: y\leq ...
0
votes
1answer
44 views

Order-preserving surjections on the Dedekind MacNeille completion

Suppose $L$ is a complete lattice, $P$ is a poset, and $f: L \to P$ is a surjective order-preserving map. If ${\bf DM}(P)$ is the Dedekind MacNeille completion of $P$, is there necessarily a ...
3
votes
2answers
223 views

Order-preserving surjection ${\mathbb N}^{\mathbb N}\to [0,\infty)$

This is kind of a continuation of a recent (closed) question. Is there an order-preserving surjective function $f:{\mathbb N}^{\mathbb N}\to [0,\infty)$ (where for $a,b\in {\mathbb N}^{\mathbb N}$ we ...
7
votes
0answers
104 views

Bound on number of nxn grids with lexicographical ordering / poset structure

Given $n\in\mathbb{N}$, consider the numbers $\{1,\ldots,n^2\}$ and a permutation $\pi\in S_{n^2}$. It induces pairs $(1,\pi(1))$, $\ldots$, $(n^2,\pi(n^2))$. Consider an $n\times n$ grid. How many ...