Questions tagged [order-theory]
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. Order theory is not about the order of a group nor the order of an element of a group or other algebraic structures.
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Is there a known well ordering of the reals?
So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My ...
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Separative Quotients, and the Induced Order
Let $P$ be some partial order.
We say that $x$ and $y$ are compatible if $\exists r\in P (r\le x \wedge r\le y)$, we denote this by $x \perp y$. Otherwise, we say that $x$ and $y$ are incompatible.
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Set Equality between Lower Bounds and and an Intersection of a Subset and Interval
The following is an example given by my professor, but there is an equality that I don't understand.
Let a partially ordered set $(S,\preceq)$ is the union of three sets such that $S=X\cup Y\cup Z$ ...
xdfm
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Showing the Inclusion is sup-continuous
I fear I over simplified the following problem:
For any partially ordered set $(A,\leq)$, let $A^* = A- \{\max A,\min A\}$ if $\max A$ and $\min A$ exist. Show the inclusion $(A^*,\leq)\...
xdmf
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For some sets $S\subseteq T\subseteq U$, when is $\inf_T S=\inf_U S$?
I've been struggling with the following problem I found for a while now:
Suppose $(T,\preceq)$ is a partially ordered subset of $(U,\preceq)$ and $S\subseteq T$. If $\inf_T S$ and $u=\inf_U S$ both ...
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Can one show that $L(E)=\langle s]$ on a partially ordered set?
This is a follow up question to one I posted earlier. I'm trying to decide that if for $(S,\preceq)$ a partially ordered set and $E\subseteq S$, one has $L(E)=\langle s]$ for some $s\in S$ iff $\inf E$...
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Does the existence of an infimum imply that the set of lower bounds of a set is totally ordered?
Say we have a partially ordered set $(S,\preceq)$, and some subset $E\subseteq S$ such that $E$ is bounded below and $\inf E$ exists. My question is, since $S$ is not totally ordered is it possible to ...
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when inf and sup of a subset achievable in itself?
I was wondering if the following is true:
In a topological space with partial order, the inf and sup for a closed subset are achievable inside the subset so that they become minimum and maximum.
Is ...
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