The order-theory tag has no usage guidance.

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### Generalizing disjointness

The following definition generalizes set-theoretic disjointess:
Definition 0. (Autonomy). Given a Lawvere theory $\mathsf{T}$, a $\mathsf{T}$-algebra $X$, and an indexed family $S$ of subalgebras ...

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### 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 ...

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### Is there a name for a partial order in which there is a countable chain which “dominates” the whole space?

Is there a name for a partial order $\preceq$ on a set $X$ with the following property: "there exists a countable set $S \subset X$ such that for all $x \in X$ there exists $y \in S$ with $x \preceq y$...

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### Complete non-isomorphic lattices with injective complete homomorphisms between them?

Are there complete lattices $L, K$ such that
$L\not\cong K$;
there are injective complete lattice homomorphisms $i:L\to K$ and $j: K\to L$
?

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### Does ${\cal Id}(L) \cong {\cal Id}(K)$ imply $L\cong K$?

For any lattice $L$ we denote the complete lattice of the ideals of $L$ by ${\cal Id}(L)$. Are there non-isomorphic lattices $L\not \cong K$ such that ${\cal Id}(L) \cong {\cal Id}(K)$?

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### Simplyfing join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_L f(S).$
Is the ...

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### 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{...

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### Lattice homomorphism from ${\cal Id}(L)$ onto $L$

For any lattice $L$ we denote the complete lattice of the ideals of $L$ by ${\cal Id}(L)$. If $L$ is complete, is there a lattice homomorphism from ${\cal Id}(L)$ onto $L$?

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### Join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_L f(S).$
Suppose $L$...

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### Incomplete lattice homomorphisms between complete lattices (2)

Let $L, K$ be complete lattices. A lattice homomorphism $f: L\to K$ is said to be incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_K f(S).$
Consider the ...

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### Incomplete lattice homomorphisms between complete lattices

Let $L, K$ be complete lattices. A lattice homomorphism $f: L\to K$ is said to be incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_K f(S).$
Suppose that $...

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### Spliting of short exact exact sequences of partially ordered groups

Consider a short exact sequence of partially ordered groups
$$0 \longrightarrow H \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} G/H \longrightarrow 0 ,$$ where $H$ is a ...

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### Embedding finite lattices into the lattice of partitions of a finite set

For any set $X$ we denote by $\text{Part}(X)$ the set of all partitions of $X$, ordered by the refinement ordering. It is well known that this is a complete lattice for all sets $X$.
Let $L$ be a ...

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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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### Finitely generated ordered monoids and noetherian subsets

(This question was asked a long time ago on MSE but got no answer so far...)
Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...

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### How long can a cycle of antichains in a finite partial order be?

Suppose that $X$ is a finite partially ordered set. Then a subset $A\subseteq X$ is said to be an antichain if there do not exist elements $a,b\in A$ with $a<b$. Let $\mathcal{A}_{X}$ be the set of ...

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### 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)&...

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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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### Can you “combine” Ord and Mon to get Cat?

Mon is the category of moniods, which can be seen as categories with one object. Ord is the category of preorders, which can be seen as categories with up to one morphism in each homset.
Is there ...

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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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### Order theory as a foundation of mathematics?

I know the followings kinds of formalization of mathematics:
based on set theory (e.g. ZFC)
based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example)
based on category ...

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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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### Counterexample on completely distributive lattices

I would like to see an example of a complete lattice $C$ which is both a frame and a dual-frame, i.e. finite meets distribute over arbitrary joins and finite joins distribute over arbitrary meets (...

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### Quasi-transitive decomposition of a transitive graph

Let $G=(V,E)$ be a simple digraph that is semi-complete (ie. there's at least one arc between each unordered pair of vertices) and quasi-transitive (ie. its complement is transitive).
Is it true that ...

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### monotonicity alike functions

assuming we have two smooth function ${f_1},{f_2}:{R^N} \to R$,
under what condition, we have
${f_1}\left( {{{\bf{x}}_1}} \right) \ge {f_1}\left( {{{\bf{x}}_2}} \right) \leftrightarrow {f_2}\...

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### Conditions for monotone function to take maximal chains to maximal chains surjectively

Suppose that $P$ and $Q$ are graded posets (with rank function $r$) and suppose that all maximal chains of $P$ and $Q$ have length $n$.
Let $f:P \to Q$ be a surjective monotone function such that $r(...

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### Is $\mathcal{P}(\omega)/fin$ with the interval topology path-connected?

Is $\mathcal{P}(\omega)/fin$ with the interval topology path-connected? (You find the definition of $\mathcal{P}(\omega)/fin$ here.)

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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 \text{...

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### Normal subgroup of a totally ordered group

A totally ordered group is a group equipped with a compatible total order, that is, $x\leq y$ and $z\leq t$ imply $x+z\leq y+t$ for all $x,y,z,t$ in the group.
Is it true that every totally ordered ...

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### Is an open map with open relative diagonal necessarily a local homeomorphism?

Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta_f : X \to X \times_Y X$ is an open embedding of locales. Does it follow that $f : X \to Y$ is a ...

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### Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space?

Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space? (You find the definition of $\mathcal{P}(\omega)/fin$ here.)
Remark: According to this, the interval topology of $\mathcal{...

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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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### 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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### Characterising subsets of the reals as ordered spaces

There are concise and elegant characterisations of the real line as a topological space and as an ordered space in the literature. I am interested in the harder case of characterising subsets of the ...

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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 $\{q\...

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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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### 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,...

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### Order-preserving image of a complete lattice

If $L$ is a complete lattice and $P$ is a poset and $f: L\to P$ is an order preserving surjective map, does this imply that $P$ is a (complete) lattice?

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### 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 $P$?...

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### Isomorphic subcategories of directed graphs and presets

For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive 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 = \{y\...

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### 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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### 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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### When do infinitesimals split in dimension groups?

Let $G$ be a dimension group (i.e. a directed, unperforated abelian group satisfying the Riesz interpolation property) with order unit $u\in G^{+}$. There is a canonical positive group homomorphism $\...

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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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### Pre-Order induced by continuous functions

I'm an newbie in category theory, but I want use it to solve a pre-order question I encountered in my research:
Let $X$ be a convex&compact subset of $\mathbb{R}^n$. $f,g: X \rightarrow [0,1]$ ...

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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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### 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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### Unique representability of bounded distributive lattices

Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space.
A poset $(P,\leq)$ is called (...

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### Minor ordering for finite graphs

Let $\mathcal{G}_{<\omega}$ be the set of graphs $G = (V,E)$ such that $V = \{0,\ldots,n\}$ for some $n \geq 0$ and $E \subseteq \mathcal{P}_2(V) = \{\{a,b\} : a,b \in V \textrm{ and } a\neq b\}$. ...