The order-theory tag has no usage guidance.

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### Prime ideals containing the finite members of ${\cal P}(\omega)$

Let ${\frak P}$ denote the collection of prime ideals containing the finite members of ${\cal P}(\omega)$, and order ${\frak P}$ by set inclusion.
What is the cardinality of ${\frak P}$, and what's ...

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### Boolean algebras and free filters generated by chains

Suppose $\mathbf{B}$ is a complete Boolean algebra with an infinite domain $B$. Suppose $\mathbf{B}$ is atomic (i.e. every element is the supremum of some set of atoms). This algebra contains the ...

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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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### Transitive reduction from a linear extension of a partial order

Is there an efficient algorithm to create a transitive reduction from a single linear extension of a given partial order?
Update: I'm aware of the time complexity of computing a transitive reduction ...

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### “Interval” terminology for (partially) ordered sets

Let $(X,\preceq)$ be a poset.
Is there a standard, generally recognised term for a set $A \subset X$ satisfying
$$ \forall x,y,z \in X, \ (x \in A \ \textrm{ and } \ z \in A \ \textrm{ ...

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### Examples of value quantales

In his paper "Quantales and continuity spaces" R. C. Flagg gives the following examples of value quantales: the lattice $\bf{2}$ of truth values with usual addition, the lattice $\mathbb{R}_{+}$ of ...

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### Properties of a specific antichain of a lattice formed by the cartesian product of finite ordered sets

Introduction
Let $X$ be a poset of all $n$-tuples, $x = (x_1, x_2, ..., x_n)$, where $0 \leq x_i \leq m_i - 1$ for $i = 1, ..., n$ together with the relation $x \prec y$ defined so that for ...

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### Relationship of ${\cal P}(\omega)/fin$ and ${\cal L}$

Define ${\cal L}$ as in this question: the set of functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ at at ...

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### 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 ...

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### A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that ...

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### Does order-preserving equal continuous? [closed]

Let $P,Q$ be posets and endow them with the interval topology $\tau_i(P)$ and $\tau_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?

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### Path-connected Hausdorff interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected and $T_2$. Does this imply that $[0,1]$ order-embeds into $P$?
(This is a follow-up ...

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### Path-connected interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected. Does this imply that $[0,1]$ order-embeds into $P$?

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### Properties of the interval topology of the lattice of functions

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 $\downarrow x = \{y\in P: ...

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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 ...

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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 ...

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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 ...

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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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### Antichains in subgroups up to group automorphism

A family of subgroups of a group G will denote a nonempty collection of subgroups closed under conjugation and further passing to subgroups.
In a Noetherian group (any ascending chain of subgroups ...

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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 ...

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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 ...

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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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### 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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### 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 ...

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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 ...

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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 ...

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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 ...

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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 ...