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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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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51 views

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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28 views

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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72 views

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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90 views

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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17 views

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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35 views

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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64 views

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 ...
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114 views

Is Wikipedia wrong about the least-upper-bound property?

Wikipedia states that the least-upper-bound property “ is a fundamental property of the real numbers and certain other ordered sets. A set $X$ has the least-upper-bound property if and only if every ...
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Let G be an abelian group, and let a∈G. For n≥1,let G[n;a] := {x∈G:x^n =a}. Show that G[n; a] is either empty or equal to αG[n] := {αg : g ∈ G[n]}… [closed]

We were given questions to study for our exam coming up. We have not covered much of this topic, so any help would be greatly appreciated! Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, ...
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Smallest Rational Number Proof [duplicate]

Can anybody help me out with solving this mathematical proof? Prove the statement “There is no smallest rational number greater than 2” by contradiction. Contradiction: There is a smallest ...
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220 views

There is no smallest rational number greater than 2

I have a problem that I am seriously stuck on. I'm not sure what to do I've seen similar proofs online with the least positive rational number but this is apparently different and I'm not sure why. ...
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Proving the principal of inclusion-exclusion using Möbius inversion

In Enumerative Combinatorics v. I, Stanley applies Möbius inversion to $B_n$, the poset of subsets of $[n]$, to show that $\forall f,g : B_n \to \mathbb{C}$, $$ g(S) = \sum_{T \subset S} f(T) \, \, ...
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Is there a bijection $f: X \to X$ such that $f(a) \succ a$ for some $a \in X$ and $f(x) \succeq x$ for all $x \in X?$

$X$ is a prewellordered set with respect to $\succeq$. The prewellordering is in general not antisymmetric relation in contrast to a well-order, i.e. $x \succeq y$ and $y \succeq x$ doesn't imply ...
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Field property proof

I have tried quite a lot of "turn and twists" but I just can`t get around it. Here it is : if x > y then x > z > y for all $x,y$ in the partially ...
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For each $x,y∈\mathbb N$, if $x≤y$ and $y≤x$, then $x=y$ [duplicate]

Proof: For each $x,y∈\mathbb{N}$, if $x≤y$ and $y≤x$, then $x=y$ By definition of order, $x≤y$ if and only if $x<y$ or $x=y$ By definition of order, $x<y$ if there exist a $K∈\mathbb{N}$ such ...
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For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$

For each $x,y,z∈\mathbb{N}$, if $x<y$ then $x+z<y+z$ I know how to solve this if it was $x=y$ then $x+z=y+z$ (which my professor said I need to do) Is there a definition that says I can change ...
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Kuratowski's closure-complement problem for the upper/lower bound functions

This is an order theory problem in the flair of Kuratowski's closure-complement problem. Let $(X,\le)\newcommand{\up}{{\uparrow}}\newcommand{\dn}{{\downarrow}}$ be a set with a reflexive order on it, ...
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Proof of Antisymmetry [duplicate]

I am stuck with proving antisymmetry under the natural numbers. Prove "For each x,y E N, if x <= y and y <= x, then x=y. I am not really sure where to start with this proof. Im assuming I will ...
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27 views

Why is this lattice modular?

I'm reading Stanley's Enumerative Combinatorics, and he says every lattice with at most six elements is modular. But what about the lattice below on the right? If $x,y$ are the two elements at the ...
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Width of Join Semilattice in $\mathbb R^n$

Suppose $S$ is a join semilattice that can be generated by $m$ generators. I would like to know an asymptotic upper bound on the width of $S$ (the maximum cardinality of all antichains in $S$) under ...
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to prove that Inf$(B) \sqsubseteq $ Inf$(A)$.

In a poset $(P, \sqsubseteq)$, and $A \subseteq B \subseteq P$. Also Infimum$(A)$ and Infimum$(B)$ exists. I need to prove that Inf$(B) \sqsubseteq $ Inf$(A)$. My Attempt. Considered $x \in A$, it ...
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If a finite nonempty set of real numbers has a maximum then it has a minimum, prove.

so as the title says. I first proved that all finite nonempty sets of real numbers have maximum, but I cannot see how from this I can prove that it also has a minimum. Any advice would be ...
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Prove every finite set of real numbers is well ordered.

so as the title says how do we prove it? I mean it is obvious but i have no idea how to do it. Without referring to general principle of mathematical induction, as that would be circular logic i ...
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Width of poset of posets

Let $P$ be the set of all labelled posets on $[n]$ ordered naturally by edge extension. What is the cardinality of the largest antichain in $P$?
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$M=\lbrace \frac{(-1)^n}{n}:n \in \mathbb N \rbrace$ not orderly countable

A non-empty set $M$ is called orderly countable if there is an surjective, strictly increasing $f: \mathbb N \to M$. Proof: $M=\left\{ \frac{(-1)^n}{n}:n \in \mathbb N \right\}$ is not orderly ...
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Partial Orders: “Minimum property” iff “Maximum property”?

Given a partially ordered set $(X,\leq)$, is it true that $$ \forall P\subseteq X:P\neq\emptyset\text{ and }P\text{ is bounded below }\to P\text{ has a minimum}\\ \updownarrow\\ \forall ...
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Show that $\mathbb N $ with the usual order is well ordered.

Show that $\mathbb N $ with the usual order is well ordered. My attempt Let $A\neq\emptyset$ a subset of $\mathbb N$. Since $A\subset \mathbb R$ is under-bounded by $0$, there is an infimum. Let ...
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Function on power set: Negativity implication leads to monotonicity

so I have a binary function $f: P \times P \to \mathbb{R}$ over some power set $P$. I also have a partial ordering $(P,≤)$ over $P$ (by set size). For this function I know that $f(a,b) ≥ 0 \implies ...
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Indecomposable representations of poset (1,1,m), m>=1

The problem says : Find indecomposable representations of poset (1,1,m) where m $\geq 1$. I've tried to find the the associated matrix problem for this poset but got nowhere. Any help would be ...
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Monotonic function which is not an order-embedding

I wrote down this definition during the lesson: Let $\langle P, \leq_P\rangle$ and $\langle Q, \leq_Q\rangle$ be two posets. A function $f: P \rightarrow Q$ is monotonic, if $x \leq_P y \implies ...