# 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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### what does ≼ or ≺ mean?

I was reading a paper about well-orderings and this came up: Suppose (E, ≤) and (F, ≼) are isomorphic well-orderings. Then there exists a unique isomorphism for (E, ≤) to (F, ≼). I've been scouring ...
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### Difficulty with exercise 24.12 in Jech

Exercise 24.12 in Jech's Set Theory (3d Millenium Edition): The lexicographical ordering $\omega \times {\omega}_1$ does not have true cofinality. True cofinality is defined (p.461) as the least ...
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### “Minimal upper bounds” in a categorical setting

It is well-known that partial orders can be seen as very simple categories (those where there is at most one morphism between every two objects). Then, the notion of "(binary) join of two elements (i....
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### Is there a common notation for $x \sqcap Yy\neq \bot$

Is there a commonly used shorthand to express the following relation: $x R y \iff X \sqcap Y \neq \bot$? That is, the greatest lower bound of the two elements is not bottom. In terms of sets, the ...
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### Well ordering of order types

Notice: in this problem I shouldn't use any properties of the ordinals, and order types are not defined as ordinals. Let $A,B$ be well ordered sets. We say that $otp(A) = otp(B)$, if $A,B$ are ...
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### Fundamental question concerning Partial/Total Orders

Often when reading various texts involving total/partial orders, I come across statements like: "...thus $\leq$ is a total order on $\mathbb{Z}$ when $a \leq b$ is given its usual meaning." Is there ...
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### Prove a fact about a mapping of complete poset to itself

Let M be a poset such that any its totally ordered subset has an upper bound: $\forall U \subseteq M$, $U$ is totally ordered $\exists m \in M: \forall u \in U \ \ u \leq m$. And let $f: M \to M$ be a ...
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### Why $(\mathbb R^*, \cdot)$ isn't an ordered group? [closed]

I don't understand why $(\mathbb R^*, \cdot)$ (group with multiplication) is not ordered ? Thank you.
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### $X \times Y$ is complete $\implies$ $X, Y$ are complete.

Apologies for the long prose. The following problems are from a set of Topology Lecture Notes by Pete L. Clark that I found fascinating and am working through. Am a little unsure about my solutions ...
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### Can I have something larger than infinite? [duplicate]

My question is "Can I have something larger than infinite?" Sometimes, we add infinite numbers into our set of numbers by simply extending our set and adding infinite numbers to it. But can't you ...
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### ordering of intervals

Suppose I have a set of N objects, {a,b,c,d,e,...}, and an NxN matrix whose values are the overlap (in length, area, volume, etc) of each pair of objects. With this matrix, can I recover the ordering ...
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### Regarding two different ordering relations.

We have 3 assumptions: 1) We have two different ordering relations $\leq_a$ and $\leq_b$. 2) Given arbitrary sets $S_1, S_2$, we know $S_1 \leq_a S_2 \rightarrow S_1 \leq_b S_2$. 3) We also ...
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### How to find the set $A \subseteq \mathbb{N}$ that $A$ is the basic set for a given divisibility relation and a given HASSE-Diagram?

How to find the set $A \subseteq \mathbb{N}$ that $A$ is the basic set for the divisibility relation: $R_A=_{def} \{(m,n) | m,n \in A \wedge m ~ \text{divides} ~ n \} \subseteq A \times A$ for the ...
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### strict partial orders that guarantee monotonic orientation functions

Given a strict partial ordered set $>$ over a set $V$. Consider its directed acyclic graph $G=(V,E)$ where $(v,u)\in E$ if $u> v$ and there is no $w$ s.t. $u> w$ and $w>v$ Consider the ...
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### Prove that a simple ordering implies a partial ordering

I am having difficulty proving theorem 55 section 3.2 page 73 of Patrick Suppes Axiomatic Set Theory. This is the theorem to be proven. (R is a simple ordering) ⇒ (R is a partial ordering). He states ...
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### Olympiad problem similar to Sperner's theorem, inspired by OMM 2 ( unproven conjecture of mine)

This problem is inspired by problem 2 here. Consider a set of cubes $F$, such that each corner $(x,y)$ of any given cube of $F$ satisfies $0\leq x,y \leq n$, and each cube has a corner with ...
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### Intersection/conjunction matrix and transforms?

I'm looking for a method to understand particular types of inclusion/intersection between objects (sets). Say I have a collection of sets, and between every pair of sets [A,B] I can derive a number ...
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### Order of $ab$ and inverses

Let $a$ and $b$ be elements of a group, with $a^2=e, b^6=e$ and $ab=b^4a.$ Find the order of $ab$ and express the inverse in each of the terms $a^mb^n$ and $b^ma^n.$ Just want to cross check my ...