# 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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### How is the Power set without the original set a partial order?

This is a past exam question. Let S = {a,b,c} Define $\mathcal{P}(S)$ as the power set of S. Consider $\mathcal{P}(S)\setminus S$ Explain how this structure illustrates the concepts of partial ...
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### Discrete Math - Hasse Diagrams

This is one of many questions of similar type I have to do for an assignment and im troubled with what to do. The question is as follows: Consider a relation R defined on the set A = {−7, −6, −5, ...
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### how to combine different partial orders (Poset)

Given two posets $\prec_A$ and $\prec_B$ where $A\neq B$ and $A\cap B\neq \emptyset$, is there any way to combine them while preserving the exact information they exhibit - namely dominance relation ( ...
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### Can a Partial Order be symmetric in addition to its properties?

For example,a relation,R defined on integers. $R = \{(a,b)\mid a^3=b^3\}$ Could this be partial order?
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### Finite ordered sets and their isomorphism

This is a slightly strange question perhaps. How many ways are there to prove that if X and Y are two finite orders (total orders) on n elements, then X and Y are isomorphic? There is a direct proof ...
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### Proof: $a<b$ and $b<c$, then also $a<c$ (for partially ordered set of $M$)

I have the following question: If $\leq$ is a partially ordered set over $M$ and $a<b$ and $b<c$, then also $a<c$ holds. I'm a little confused, because I know that a partially ordered set ...
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### How many ordinals can we cram into $\mathbb{R}_+$, respecting order?

I've been pondering the following question. How can we measure the amount of "space" above an element $p$ in a partially ordered set $P$? One way would be to try to cram the elements of ...
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### Kinda well-orderedness.

Its not the case that $\mathbb{Z}$ is well-ordered (under the usual order). However, observe that Every non-empty subset of $\mathbb{Z}$ that is bounded below has a least element, and Every ...
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### Drawing lattices

I would like to be able to draw the lattices of subgroups of certain groups. I thought it would be easy when I already know the structure, but I've never done it before, and it turns out it's harder ...
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### Why should the union of such a family of sets have cardinality $\aleph_1?$

Let $\mathcal A$ be a family of infinite countable sets which is linearly ordered by inclusion and such that $\bigcup\mathcal A$ is uncountable. I have to prove that $|\bigcup\mathcal A|=\aleph_1.$ I ...
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### Proof for a creating a partition of a countable set using chains in partial orders.

Definition: A partition of a set $A$ is a set of nonempty subsets of $A$ called the blocks of the partition, such that  every element of $A$ is in some block, and  if $B$ and $B'$ are different ...
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### Preference Relation and Utility Function - Problem with inductive proof

I have a problem with an inductive proof of the following result. Theorem: If $X$ is a finite set, a binary relation $\succ$ is a preference relation iff there exist a function $u:X\rightarrow R$ ...
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### What exactly is a lattice? And can somebody give an example of something that is not one?

Looking at the (very brief) definition in my textbook with no examples, I have the following: A poset $(A,\preceq)$ in which every two elements have a greatest lower bound in $A$ and a least upper ...
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### Partition of lattice into symmetric chains

Let $C_1 \subsetneq C_2 \subsetneq C_3 \subsetneq \cdots \subsetneq C_k$ be a chain in the subset lattice $(2^{[n]},\subseteq)$. A chain is considered symmetric if $|C_1| + |C_k| = n$ and ...
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### Trichotomy of well ordering

I'm trying to prove trichotomy of well ordering and I have a question concerning the proof of the following fact: given two well ordered set $(A,\triangleleft_A),(B,\triangleleft_B )$, then the ...
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### Does there exist an ordered ring, with $\mathbb{Z}$ as an ordered subring, such that some ring of p-adic integers can be formed as a quotient ring?

I'm not sure if this might instead be MathOverflow material, but I'll give it a shot here first anyway. Does any ring $R$ exist that satisfies the following properties? $R$ is a totally ordered, ...
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### Partial order Proof

Suppose that $<$ is a partial order on a set $A$. Prove that there exists a linear order $<'$ on $A$ such that for all $x,y \in A$, if $x < y$ then $x <' y$. Please someone give me ...
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### A problem about compact order topological space

This problem is from terrytao.wordpress.com/books/analysis-ii,Errata to the second edition (hardcover),P.390,Exercise 13.5.8 .I have difficulty proving this,appreciate any help! Show that there ...
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### The proper subset relation and strict partial order

The proper subset relation, $\subset$, defines a strict partial order on the subsets of $[1,6]$, that is $pow[1,6]$. (a) What is the size of a maximal chain in this partial order? Describe one. (b) ...
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### Procedure for Max Function with identical random variables

I am studying for the P/1 actuarial exam, and I keep encountering one type of problem which I can't seem to solve and can't seem to find any generalized explanation for how to do it in my textbooks. ...
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### Total orders making a family of functions non-decreasing

Suppose $S$ is a finite set and we are given a family of functions $(f_i:S\to S)_{i\in I}$. When can we find a total order $<$ on $S$ such that all the functions $f_i$ are nondecreasing? I am ...
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### If every subset of a total order $(P,\leq)$ is isomorphic to an initial segment of $(P,\leq)$, then is $P$ a well-order?

I have to proof that "If every subset of a total order $(P,\leq)$ is isomorphic to an initial segment of $(P,\leq)$, then $P$ is a well-order". I already got the proof of the converse and this ...
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### A new(?) partial order on the set of continuous maps

Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a ...
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### Ordinal interpretation of Friedman's $n$?

I heard that Kruskal's tree theorem can be turned into a finite form that creates an extremely fast growing function because ordinals could be encoded into trees. On this wiki page it mentions that ...
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### Confused about preservation of well-quasi-orders

I don't understand preservation of well quasi-orders, here is what I have thought about: Definition A binary relation $R$ on $X$ is a quasi-order when it is reflexive and transitive. Definition A ...
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### inf and min in partial order

Assume $P$ is a partial order. Does it holds that $\inf (\inf(a,b), \inf(c,d)) = \inf(a,b,c,d)$? My guess is that it holds. But I don't know how to check it. Also what if instead of $\inf$ I use ...
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### Constructing an order-preserving function between two strict p.o.s that is not one-to-one.

Out of Winfried Just and Martin Weese's Set theory book: Give an example of strict partial orders $\langle X,<_x\rangle$ and $\langle Y,<_y\rangle$ and of a function $\tau : X \to Y$ that is ...
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### From point-wise to essential supremum of a set of real-valued measurable functions

I want to prove some property about essential suprema and I think I can show them for the pointwise supremum $\sup S$. The problem is, that the sets involved are uncountable and thus, the point-wise ...
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### What are ideals in a complete lattice?

If I have a complete lattice $L$, what conditions do I need for $I\subseteq L$ to be an ideal? In a general lattice the conditions are: $I$ is a lower set; $I$ is closed under (finite) joins. For ...
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I have these two definitions, about minimals and minimums in an order relation: $b$ is minimal in $B<=>¬\exists:(x \in B \land x \prec b)$ and also equivalent to $\forall x,[(x \in B \land x ... 1answer 123 views ### Total Ordering and relation Consider the relation$<$on$\mathbb{Q}$defined by:$(m, n) < (j, k) \iff jn-mk \in \mathbb{N}.$Where$m, j \in \mathbb{N}$and$n, k\in\mathbb{Z}$I want to show that$<$is a total ... 1answer 106 views ### Theorem unique bijection between two well-ordered sets satisfying certain conditions I have some problems understanding the proof of the following theorem. Given two well-ordered sets$(X, \le _X), \ (Y, \le _Y)$there exists exactly one partial function$f(X, Y)$such that: 1)$f: \ ...
Write $E_k = (\{0, \ldots, k-1\}, \leq)$ the total order on $k$ elements. If $(A, \leq_A)$ and $(B, \leq_B)$ are two posets, the product poset is $(A \times B, \leq_{A \times B})$ with the order ...