# Tagged Questions

Order theory deals with properties of orders, usually partial orders or quasi orders but not only those. Questions about properties of orders, general or particular, may fit into this category, as well as questions about properties of subsets and elements of an ordered set.

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### Neighborhood chain in $\mathbb{R}^\mathbb{R}$

Edited after SamM's comment: Consider the topological space $\mathbb{R}^\mathbb{R}$, with the usual topology. Pick a point $x \in \mathbb{R}^\mathbb{R}$ and a neighborhood $V = V_0$ of $x$. I wish to ...
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### Why aren't there uncountably many disjoint open intervals of $\mathbb{R}$?

I know that this can't be true given that $\mathbb{R}$ is separable, but I'm having a hard time coming to grips with why this is, exactly. In particular, can't I just take some uncountable strictly ...
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### Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is ...
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### Lattices and Boolean algebra

I have read in a text book that the set of natural numbers form a lattice under divisibility. How can it possibe, since there is no upper bound and therefore a Sup of the set?
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### Prove every finite lattice has a greatest element - without induction

I have to prove that every finite lattice (L, ≤) has a greatest element. I have seen a lot of proofs proving this by using induction, however, I have to prove it without induction since our ...
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### Hasse diagrams for sorts of size 3 and 4?

Does anyone know what does the following mean? I understand that a Hasse diagram represents a given partial order but I don't seem to get this example. Below is a Hasse diagrams for sorts of size 3 ...
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### Adding relations to a partial order

Let $(X,\leq)$ be a poset. For every $A\subseteq X$, $a \in A$ is called isolated iff there are only finite number of elements in $A$ that can be compared to it. A continuous (probably not the best ...
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### Example of non-way-below Auxiliary Relation in a Poset.

As the title have suggested, I am looking for an example of an auxiliary relation in a poset that is not the way-below relation. Any suggestion of books/paper to read is appreciated. Thanks. Edit: ...
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### Higher order partial order

I'm trying to find an effective and making sense way (for programming purposes, but also from math interest) to order partial orders, means, given a set and many partial orders (trees) over it. How ...
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### Free cocompletion of thin categories

Let $C$ be a thin category. Is there a cocomplete thin category $C\to C'$ such that for every functor $C\to D$ into a cocomplete thin category there is a unique, up to isomorphism, cocontinous functor ...
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### Partially ordering finite graphs

What are some interesting partial orders on the set of all finite graphs (identified up to isomorphism), apart from the usual (induced) subgraph relation and the (topological) minor relation, and why ...
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### Given a partial order $R$ on the set $A$, prove that there exists a total order $\leq$ on $A$ such that $R \subseteq {\le}$

Let $A$ be a finite set, and $\langle A,R\rangle$ be partially ordered. $\textbf{Prove}$ that there exists a total order $\leq$ over $A$ such that $R \subseteq {\le}$. $\textbf{My Attempt:}$ ...
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### Filters and antichains in preorders

In a preorder $\mathbb{P}$ (reflexive and transitive), what means to say that a subset $C\subset\mathbb{P}$ is not contained in any filter of $\mathbb{P}$? For example, it is obvious that if $C$ ...
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### How well-behaved can well-orders on $\mathbb{R}$ be?

Well-orders on $\mathbb{R}$ are interesting because they witness the "alephness" of the cardinality of the continuum, and they're therefore connected to the continuum problem. It seems reasonable, ...
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### Why do people accept the axiom of choice given the well ordering principle?

We know without any doubt that the axiom of choice implies (in fact is equivalent to) the well ordering principle. The well ordering principle can't be true! If we take the open interval $(0,1)$ for ...
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### When is a join semilattice not a meet semilattice?

When would a join semilattice not also be a meet semilattice (and vice versa)? I can't think of any way. An example or two would probably be very helpful.
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### Can every poset be toplogically sorted?

A partial ordering $\preceq'$ on $S$ is said to be compatible with another partial ordering $\preceq$ if for all $a,b,\in S$, $$a \preceq b \Rightarrow a \preceq' b$$ Given a poset $(S, \preceq)$, ...
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### Is BPIT equivalent to some ordering principle?

Working in $\mathsf{ZF}$, is $\mathsf{BPIT}$ (Boolean Prime Ideal Theorem) equivalent to some statement of the form "every set can be ***ly ordered"? I know that $\mathsf{BPIT}$ implies that every set ...
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### Product of directed partial orders

Is a product poset (with componentwise order) of nonempty posets a dcpo if and only if each multiplier is a dcpo? (for both binary and arbitrary products)