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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Example of discrete set [closed]

Please I need examples of discrete set and non discrete set... am a little confused over this expression. I am thinking of discrete as a finite set but found from another article that, its not ...
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Prove 1 is not the largest integer? [duplicate]

This proof looks extremely flawed, but I'm new to proofs so I'm not completely sure what is allowed and what isn't. Here it is: Let $n$ be the largest positive integer. Then $n$ must be $\geq 1$. ...
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Is there a sequence of with every other sequence as a subsequence?

Let's say we have a set $S$. Is there a sequence $u:\mathbb N \to S$ such that every other sequence $v$ is a subsequence of $u$? Here is what I have so far: If $S=\emptyset$, then no (there are no ...
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History and motivation for Continuous Posets.

Can anyone point to me resources that motivates discusses the history and development of continuous posets? I would really like to know what 'concrete' objects that were studied that gives rise to ...
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Embedding tuples of natural numbers into real numbers

I'm looking for a pointer to common techniques to map tuples of natural numbers into real or rational numbers such that the ordering is preserved (I assume tuples are ordered lexicographically). I'm ...
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Interpreting N vector function

For $\mathcal{N} = \mathcal{N} \bigcup\{ \bot_{\mathcal{N}} , \top_{\mathcal{N}} \} $ and $\mathcal{N}^\vec{} = \mathcal{N} \vec{} \mathcal{N} $ (monotonic function over $ \mathcal{N} $), how ...
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Partial orders that can be realized by prime ideals of commutative rings

Is there a general criterion for which partial orders can be realized by the prime ideals of commutative rings (like we have for topological spaces - https://en.wikipedia.org/wiki/Spectral_space)? ...
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Does there exist a poset whose “quasiwidth” strictly exceeds its width?

Let $P$ denote a poset. Then the width of $P$ is defined as the supremum of the set of all cardinal numbers $\kappa$ such that $P$ has an antichain of cardinality $\kappa$. By the quasiwidth of $P$, ...
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Are ($Q$, $\leq$) and ($Q \times Q$, $\leq _e$) isomorphic? [duplicate]

I can't really tell if ($Q$, $\leq$)$\cong$($Q \times Q$, $\leq _e$), where $\leq_e$ denotes the left lexicographic order. Neither have a last/first element, both are dense and have the same ...
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Find a poset with no infinite antichains and that can only be covered by infinitely many chains.

So far I'm struggling to find such a poset. I first considered $(\mathbb{Z}, \leq )$ because it doesn't have any antichains so there aren't any infinite ones. The problem is that whilst the chain ...
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Can every countable Boolean algebra be embedded into $\mathcal{P}(\mathbb{N})$?

Can every countable Boolean algebra be embedded into $\mathcal{P}(\mathbb{N})$? And if so, is the same true for countable semi-lattices?
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Interpreting functions for poset

I'm stuck at a question with the following expression of poset $(\mathcal{N}, \sqsubseteq)$: $$\forall x, y \in \mathcal{N}: x \sqsubseteq y \mbox{ iff } (x = y \vee x = \bot_{\mathcal{N}} \vee y = \...
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The lattice of partial partitions

In perusing A kind of Eulerian numbers connected to Whitney numbers of Dowling lattices I have gotten confused over what seems a very elementary point. The beginning of section 2 of the paper is ...
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Interesting orderings open sets of a topology

Let X be a set of points an $\mathcal{Q}$ be a partition on X. The intuition I want to model is that X is a set of worlds considered possible by an agent and $\mathcal{Q}$ is a question whose ...
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49 views

Different Zorn's lemma statements

Given a chain-complete poset $P$, every $x\in P$ lies below some maximal element. Every inductive poset has enough maximal elements a maximal element. Chain-complete means every chain has a least ...
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Proving a total-order is a well-order if and only if every initial segment is determined by an element

I managed to prove the $\longrightarrow$ part, but I'm not entirely sure how to prove the second part. I can assume by contradiction that the total order $(X, \leq)$ is not a well-order, which means ...
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Algebraic lattices and distributivity over joins of upward directed sets

I am reading Burris & Sankappanavar, Chapter 1 on lattices, and I am doing Exercise 6 in Section §4: If $L$ is an algebraic lattice and $D$ a subset of $L$ such that for each $d_1$, $d_2 \in D$ ...
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Alternate definitions of width (of a partial order) without Choice?

Say an antichain of a poset $P$ is a set of pairwise incomparable elements of $P.$ Typically, the width of a partial order is defined to be the supremum of the cardinalities of antichains of $P.$ When ...
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Ordered sets $\langle \mathbb{N} \times \mathbb{Q}, \le_{lex} \rangle$ and $\langle \mathbb{Q} \times \mathbb{N}, \le_{lex} \rangle$ not isomorphic

I'm doing this exercise: Prove that ordered sets $\langle \mathbb{N} \times \mathbb{Q}, \le_{lex} \rangle$ and $\langle \mathbb{Q} \times \mathbb{N}, \le_{lex} \rangle$ are not isomorphic ($\le_{lex}$ ...
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Question about partially ordered sets and how to order them?

If I had a partially ordered set (E, ≤) where E = {1,2,3}, how would I order it? In other words, if I wanted to show the set, would I need to conform to the definition of ≤? So would it be {(1,2), (1,...
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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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Define well-order on finite subsets of $\mathbb{N}$ [closed]

Define a well-order $\le$ in $P_\text{fin}(\mathbb{N})$ - (it's the set of all finite subsets of $\mathbb{N}$, the natural numbers) that: $ A, B \in P_\text{fin}(N) (A \subseteq B \rightarrow A \le B ...
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Examples of adjoint order morphisms

Given posets $(X, \leq)$, and $(Y, \leq)$, an order morphism is a map $f : X \to Y$ such that $x \leq y$ implies $f(x) \leq f(y)$. Given two order morphism $f: X \to Y$ and $g: Y \to X$, we say that $...
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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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A property of product order

Let $\mathfrak{A}$ be a poset, let $a\in\mathfrak{A}$. By definition $$\star a = \{ x\in\mathfrak{A} \mid \text{there exists non-least } y\in\mathfrak{A} \text{ such that } y\le a \text{ and } y\le x\}...
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On the order of natural functions {f:N→N}

Define a partial order on natural-valued functions (or sequences, depends on how you see it): $f<g$ iff $\exists x:\forall n(n>x\rightarrow f(n)<g(n))$. Intuitively, $f<g$ if $g$ ...
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Can every non-empty set satisfying the axioms of $\sf{ZF}$ be totally ordered?

Let us first propose the following axiom, Axiom of Ordering $(\sf{AO})$. If $S$ be a non-empty set. Let $a,b\in S$. Then the set $\{a,b\}$ can be totally ordered. Now let us consider $\sf{ZFO}$ ...
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closed monoidal posets

Let $X$ be a set, regarded as discrete category. if $X$ has structure of closed monoidal category $(X,\cdot,e)$, then it is easy to show that $X$ is a group: since all the morphisms are identities, ...
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Which of the sets are well ordered? Which ones are isomorphic?

I'm new to StackExchange and I'd like to ask you for help. I have been trying to solve this exercise: $$A = \left\{3 - \frac{1}{2n} : n \in \mathbb{N} - \left\{ 0 \right\} \right\}$$ $$ B = \left\{ \...
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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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What is a partial order? Are $<$ and $\le$ partial orders?

According to Jech's set theory, a partial ordering (of $P$) is: A binary relation $<\subseteq P\times P$ where we have: $\forall p\in P[(p,p)\notin <]$ (which we may say as: $\forall ...
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What is the difference between an Ordered Set and a Completely Ordered Set?

When is a set called an Ordered Set and when is it called a Completely Ordered Set?
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What is the name of this construction of an induced order?

Let $<_1$ be a total order on a non-empty set $A$. Consider the set $\mathscr{J}(A)$ given by $$\mathscr{J}(A) = \bigsqcup_{n=1}^\infty A^n$$ ...where $\sqcup$ denotes "disjoint union" (or $\...
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$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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“General position” in a poset

Does anyone have a reference for this notion of "general position" in a poset: A set $S$ in a poset $(X,\le)$ is said to be in general position if for any $A,B\subseteq S$, $\{x:\forall y\in B,y\...
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$f'(x)\geq 0$ only, but still strictly increasing

It is a quick corollary of the Mean Value Theorem that for a real differentiable function $f:(a,b)\rightarrow\mathbb R$ that if $f'(x)\geq 0$ for all $x\in (a,b)$ then $f$ is (weakly) increasing if $...
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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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A question on proof of the Zermelo's theorem “Every set is well-orderable.”

There exists some proof for the theorem. Some of them use Transfinite Recursion. Some of them use same argument with the following. Proof:(copied from proofwiki) "Let $S$ be a set. Let $\mathcal{P}(...