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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difference between maximal element and greatest element

I know that it's very elementary question but I still don't fully understand difference between maximal element and greatest element. If it's possible, please explain to me this difference with some ...
18
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6answers
534 views

Embedding ordinals in $\mathbb{Q}$

All countable ordinals are embeddable in $\mathbb{Q}$. For "small" countable ordinals, it is simple to do this explicitly. $\omega$ is trivial, $\omega+1$ can be e.g. done as $\{\frac{n}{n+1}:n\in ...
11
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2answers
389 views

Any good decomposition theorems for total orders?

I'm learning about set theory and partial orders, well orders, total orders, etc. I'm curious to know of any good decomposition theorems that apply to all (or very general types of) total orders. ...
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2answers
757 views

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, ...
11
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3answers
566 views

Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$

In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization: A lattice is distributive if and only if ...
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3answers
351 views

Without appealing to choice, can we prove that if $X$ is well-orderable, then so too is $2^X$?

Without appealing to the axiom of choice, it can be shown that (Proposition:) if $X$ is well-orderable, then $2^X$ is totally-orderable. Question: can we show the stronger result that if $X$ is ...
3
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1answer
116 views

Problem with “tree” definitions

In my studies of choice principles, I've encountered the concept of a tree several times. Frustratingly, no two of the sources I've been working with define it in quite the same way! Notation: Given ...
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2answers
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limsup and liminf of a sequence of subsets of a set

I am confused when reading Wikipedia's article on limsup and liminf of a sequence of subsets of a set $X$. It says there are two different ways to define them, but first gives what is common for ...
9
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2answers
649 views

The p-adic numbers as an ordered group

So I understand that there is no order on the field of p-adic numbers $\mathbb{Q_p}$ that makes it into an ordered field (i.e.) compatible with both addition and multiplication. Now, from the ...
9
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2answers
1k views

Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$

I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227. A partial ordering $R$ is said to be dense iff whenever $xRz$, ...
6
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1answer
466 views

Totally ordering the power set of a well ordered set.

Let's say I take a set $S$, where $S$ can be well ordered. From what I understand, one can use that well ordering to totally order $\mathscr{P}(S)$. How does a body actually use the well ordering of ...
11
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2answers
810 views

Every partial order can be extended to a linear ordering

How do I show that every partial order can be extended to a linear ordering? I think that I manage to prove that claim for finite set, how can I prove it for infinite set? Thank you.
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1answer
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limsup and liminf of a sequence of points in a set

My ways to define/write limsup and liminf of a sequence of points in a set $X$: They come from what I have understood. If you have other ways of understanding, really appreciate if you can reply ...
4
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3answers
220 views

Countable ordinals are embeddable in the rationals $\Bbb Q$ — proofs and their use of AC

Yesterday, Asaf Karagila's answer to my question sparked an extensive discussion on ways of proving that all countable ordinals are embeddable in $\Bbb Q$, and whether particular solutions to this use ...
3
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2answers
135 views

On Tarski-Knaster theorem

Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion. If $F: P(X) \to P(X)$ is monotone (order preserving), then $F$ has a fixed point. How ...
8
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1answer
133 views

Do there exist totally ordered sets with the 'distinct order type' property that are not well-ordered?

Define that the order type of an element $x$ in a totally ordered set $X$ is the order type of $\{w \in X\mid w < x\}$. Under this definition, distinct elements of a well-ordered set have distinct ...
2
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2answers
452 views

Infinum & Supremum: An Analysis on Relatedness

$\require{color}$ Question: I need some help in proving that $\color{green}{\text{if $k\geq 0$, then $\sup (kS) = k\sup(S)$ and $\inf (kS) = k\inf (S)$}}$, and also that $\color{red}{\text{if ...
2
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2answers
245 views

Bourbaki-Witt fixed point theorem: two questions

Consider the following theorem: Let $f\colon E \to E$ have the propery that $f(x)\geq x$, where $(E,\leq)$ is a non-void partially ordered set with the property that every totally ordered subset of ...
8
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2answers
237 views

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 ...
5
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1answer
131 views

If $X$ is an order topology and $Y \subset X$ is closed, do the subspace topology and order topology on $Y$ coincide?

Let $(X,<)$ be a linearly ordered set and give $X$ its order topology. Suppose that $Y \subset X$. There are 2 sensible ways to topologise $Y$: View $Y$ as a subspace of $X$ and give it the ...
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3answers
459 views

How to define a well-order on $\mathbb R$?

I would like to define a well-order on $\mathbb R$. My first thought was, of course, to use $\leq$. Unfortunately, the result isn't well-founded, since $(-\infty,0)$ is an example of a subset that ...
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4answers
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Simplest Example of a Poset that is not a Lattice

A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a ...
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1answer
257 views

Question of an isomorphism of $\epsilon_ 0$ and a subset of the rationals.

I don't know if this question is appropriated for this site. Anyway, I'm searching for an isomorphism of order $f:K \longrightarrow \epsilon_o $, such that $(K, \leq)$ is a subset(proper or not) of ...
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1answer
109 views

preorders induced by continuous functions to the reals

Any function to a (total) order induces a (total) preorder on its domain. What can be said about the total preorders induced by a continuous function from a "nice" topological space (for instance, a ...
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2answers
451 views

A question regarding the Continuum Hypothesis (Revised)

Given that the order type of the reals $(\mathbb R,<)$ has no definable points, can it be proven that $|\mathbb R|= \aleph_1$ by virtue of the fact that every uncountable proper subset $S$ of ...
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1answer
375 views

Rudin Theorem 1.11

After spending a few hours trying to understand Theorem 1.11 in Rudin's Principles of Mathematical Analysis, I still don't follow the proof. 1.11 Theorem Suppose $S$ is an ordered set with the ...
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1answer
256 views

Cross Product of Partial Orders

im going to have a similar questions on my test tomorrow. I am really stuck on this problem. I don't know how to start. Any sort of help will be appreciated. Thank you Suppose that (L1;≤_1) and ...
6
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1answer
75 views

Are there any nontrivial doubly-well-ordered sets?

A set $A$ is well-ordered (by an ordering $R$) if there is an $R$-minimal element in every nonempty subset of $A$. Call $A$ doubly well-ordered (by $R$) if $R$ well-orders $A$ and $R^c$ (converse) ...
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2answers
155 views

Showing there is only one isomorphism between well ordered sets using transfinite induction

I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do ...
4
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2answers
127 views

$<$ in a preorder

The author of the book I am studying defines $<$ for a poset as If $x, y \in X$, where $X$ is a poset, then we shall write $x < y$ to mean that $x \le y$ and $x \ne y$. From this, I can ...
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1answer
122 views

If $A$ is well ordered and $B$ is well ordered then $A\times B$ with the lexicographic order is well ordered

I am trying to prove that: Given that $A,B$ are well ordered sets, also $A \times B$ is a well ordered set. (A set $A$ is well ordered if a linear order is defined on it, and every non empty subset ...
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3answers
108 views

Equal elements vs isomorphic elements in a preoder

While introducing preorders, Roy L. Crole in Cateories for Types states If $x \leq y$ and $y \leq x$ then we shall write $x \cong y$ and say that $x$ and $y$ are isomorphic elements. I'm trying ...
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1answer
102 views

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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3answers
317 views

How many infima and suprema can there be?

The Wikipedia definitions of infimum and supremum include the words "greatest" and "lowest", implying that there's at most one infimum and one supremum for any given subset. But in the empty set ...
0
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2answers
123 views

Recurrence Relation Solving Problem

Can anyone help me in solving this complex recurrence in detail? $T(n)=n + \sum\limits_{k-1}^n [T(n-k)+T(k)] $ $T(1) = 1$. We want to calculate order of T. I'm confused by using recursion tree ...
0
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1answer
109 views

Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing singleton sets?

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
0
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1answer
26 views

Are there necessary and sufficient conditions for Krein-Milman type conclusions?

This, the third of three self-answered questions, contains a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The first question is here. The second question is here. ...
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1answer
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Conditions for augmenting a collection of sets so that the new sets are small but the hull is large?

This is the second of three self-answered questions which will culminate in a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The first question is here. The third ...
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1answer
29 views

Are there necessary and sufficient conditions so that every element in a partially ordered set is either the least element or in the upset of an atom?

This is the first of three self-answered questions which will culminate in a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The second question is here. The third ...
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1answer
150 views

The preorder of countable order types

Consider the set $\mathcal{O}$ of order types corresponding to all posets of cardinality at most $\aleph_0$. The set $\mathcal{O}$ is a preorder under embeddability of its elements (note that some ...
10
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1answer
347 views

Linearly ordered sets “somewhat similar” to $\mathbb{Q}$

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{\eta} + \pmb{1}$ - order type of $\mathbb{Q}\cap(0, 1]$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. Let's say that a linear ...
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2answers
329 views

Is $(\pmb{1} + \pmb{\eta})\cdot\pmb{\omega_1} = \pmb{1} + \pmb{\eta}\cdot\pmb{\omega_1}$?

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{1}$ - order type of a singleton set. $\pmb{\omega_0}$ - order type of $\mathbb{N}$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. ...
10
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1answer
166 views

Is $\pmb{\eta}\cdot\pmb{\omega_1} = (\pmb{\eta} + \pmb{1})\cdot\pmb{\omega_1}$?

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{1}$ - order type of a singleton set. $\pmb{\omega_0}$ - order type of $\mathbb{N}$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. ...
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3answers
247 views

Help me get hyped about lattices

I own a small book on Lattice theory published by Dover. Unfortunately, since I bought it almost a year ago, I have gotten nowhere in its study. What I am asking for are freely available papers that ...
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3answers
921 views

Isomorphisms: preserve structure, operation, or order?

Everyone always says that isomorphisms preserve structure... but given the (multiple) definitions of isomorphism, I fail to see how the definitions equate with the intuitive meaning, which is that two ...
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2answers
265 views

Is every suborder of $\mathbb{R}$ homeomorphic to some subspace of $\mathbb{R}$?

Let $X \subset \mathbb{R}$ and give $X$ its order topology. (When) is it true that $X$ is homeomorphic to some subspace of $Y \subset \mathbb{R}$? Example: Let $X = [0,1) \cup \{2\} \cup (3,4]$, ...
10
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3answers
2k views

What do curly less than sign mean?

I am reading "Convex Optimization" by Stephen Boyd. He is using a curved greater than and curved less than equal to signs. $f(x^*) \succcurlyeq \alpha$ or $f(x*) \preccurlyeq \alpha$ Can someone ...
10
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0answers
471 views

How to find a total order with constrained comparisons

There are 25 horses with different speeds. My goal is to rank all of them, by using only runs with 5 horses, and taking partial rankings. How many runs do I need, at minumum, to complete my task? As ...
8
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1answer
199 views

Is this a characterization of well-orders?

While grading some papers and thinking about a question related to well-orders (in particular, pointing a mistake in a solution), I came to think of a reasonable characterization for well-orders. I ...
6
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2answers
118 views

Every preorder is a topological space

Let $(X,\leq)$ be a preorder, i.e. $\leq$ is a reflexive and transitive binary relation on $X$. I want to show that $\leq$ induces a topological structure on $X$. Hence I need to specify when a subset ...