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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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 ...
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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 $A,B\subseteq S$, $\{x:\forall y\in ...
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$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 ...
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A group with an element of finite order and another element of infinite order?

I have a (probably) stupid question. I'm in an intro to group theory class and I am trying to better grasp the concept of element order. My question is if I have a group and it has an element of order ...
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order relations & doubts

It is given a relation $R=\left \{(1,1); (2,2); (3,3); (4,4); (5,5); (3,5); (4,3); (4,2); (4,5); (2,5)\right \}$ for a set $A=\left \{ 1,2,3,4,5\right \}$. I found this relation is reflexive, ...
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Definitions Continuity Using the Order (Landau) symbolism.

Is an $o(1)$ function always continuous?
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Prove that every discrete poset is projective.

Is this proof correct (assuming the Axiom of Choice)? $e:P\longrightarrow Q$ epic, then $e$ is a surjection (Proof that the epis among posets are the surjections). For any arrow, $f:A \longrightarrow ...
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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 ...
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Abelian Group G and it's order (Let G be an abelian group, and let a∈G)

I already asked this question. However, it was closed off. I now have inserted by attempt at the solution: Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, let $G[n:a] := \{x\in G:x^n ...
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Abelian Groups and orders

I came across this question, and was wondering what the notation G[n : a] means. Is x the generator? Any comments/explanations will be of great help. The question I found was: Let $G$ be an abelian ...
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Definitions On Landau Notation (Big O and little o)

What are the definitions on Big O and little o for when $x \in R^m$ approaches $s\in R^m$ And not $x$ going to infinity?
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Proving that restrictions of partial orders are partial orders

Prove: A set has a partial-order relation $R$ on it. $P$ is a subset of this set. Prove that the restriction of $R$ to $P$ is itself a partial-order relation. Assume that this relation, $T$, ...
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A function is monotonically increasing if for each i, j ∈ ℤ m, i < j ⇒ f(i) ≤ f(j).

Heres the whole question but I'm completely lost on even how to start or what to do. If anyone wouldn't mind showing me how to do part a I think I might be able to nudge my way through the rest. For ...
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How many chains are there in a finite power set?

Let $A$ be a finite set with $n$ elements. How many chains are there in $\mathcal P(A)$ -- that is, how many different subsets of $\mathcal P(A)$ are totally ordered by inclusion? It's easy enough to ...
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On the relation of Completeness Axiom of real numbers and Well Ordering Axiom

In my abstract algebra book one of the first facts stated is the Well Ordering Principle: (*) Every non-empty set of positive integers has a smallest member. In real analysis on the other hand one ...
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Why isn't there a total order of $\cal P(\Bbb R)$?

I have heard that, in some models of ZF, $\cal P(\Bbb R)$ has no total order. How could one prove this? I already know that $\Bbb R$ isn't necessarily well-ordered. I'm guessing that one could reduce ...
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Inclusion in posets

In this problem we work in the poset of naturals with the divisibility order, i.e, $k \leq n$ if and only if $k \ \vert \ n$. For $i \in \mathbb{N}$ fixed let $A_i:= \big\lbrace p_i^{\alpha_i} : ...
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How big the maximal decrease in consecutive elements of a sequence?

Consider a sequence $(s_1, ..., s_k)$, and we have it sorted in decreasing order $(\tilde{s}_1, ..., \tilde{s}_k)= (\sigma(s_1), ..., \sigma(s_k))$. Define $k_{\max} = \max \left( \max_i \left( s_i ...
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Why is the union of two closed sets again closed in the Scott topology?

I want to figure out why the Scott topology really forms a topology. In particular, I want to find out why the union of two closed sets is again closed. So say we have a partial order $P = (P, ≤)$. A ...
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The $n$-cells of a modular lattice are antichains

The terminology "$n$-cell" is made up by me, and I'd love to hear if this has an official name. Given a poset $(X,\le)$, define the set $A_n$ of $n$-cells of $X$ recursively as follows: An ...
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Name of the class of maps between posets such that $f(x)\le f(y)\implies x\le y$

Is there a name for such functions $f:\mathfrak{A}\rightarrow\mathfrak{B}$ from a poset $\mathfrak{A}$ to a poset $\mathfrak{B}$ that $$f(x)\le f(y)\implies x\le y$$ (for every $x,y\in\mathfrak{A}$)?
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Prove that this is a partial order

I am reading Charles Pinter's "Set Theory" and I found an exercise which I can't resolve. Maybe someone can help me. It says: We will consider pairs $(B,G)$ where $B \subseteq A$, and $G$ is an ...
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If not $b<a$, then $a\leq b$

In trying to prove the following inequality $\neg(a\leq b)\Longrightarrow b<a$ i could produce the following indirect proof: Let $\neg(a\leq b)$. Let $\neg(b<a)$ But ...
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Is there a name for a total order with a zero element?

Is there a name for a total order equipped with a zero/bottom element? (And not necessarily a unit/top element.)
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Can an uncountable set be a “chain”?

In the Wikipedia page about total orders, it's stated that a "chain" is a synonym for a totally ordered set. But this makes no sense to me, since "chain" seems to suggest countability, and yet the ...
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Equivalence of two definitions of complete distributivity

I would like to know if the following alternative definitions of complete distributivity are equivalent. Let's begin with defining choice functions: For any set $S$ and $U\in ...
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Question about proof of “every finite poset has a maximal element”.

I have seen various proofs (see, for example, here: http://math.stackexchange.com/a/436074/264885) where we're saying if $a$ is not a maximal element, then there necessarily is some $b$ such that $a ...
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Is the width of a poset well-defined?

According to the Wikipedia definition (current revision), the width of a poset is the cardinality of any maximum antichain, where "maximum antichain" here means an antichain of maximal cardinality. ...
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Connections between Posets and WQO's

Here is the question that I posted on the Mathematics Chat Room that I was unable to find an answer to: Question: Under what conditions/properties is a poset ever a wqo (well-quasi-order)? Can we ...