# Tagged Questions

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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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### 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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### 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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### 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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### 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 $\mathbb{R}^n$. $f,g: X \rightarrow [0,1]$ ...
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 $P$ ...