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.

learn more… | top users | synonyms

0
votes
2answers
273 views

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 ...
0
votes
0answers
54 views

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 ...
1
vote
1answer
90 views

“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....
0
votes
0answers
18 views

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 ...
-1
votes
3answers
54 views

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 ...
0
votes
0answers
26 views

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 $...
1
vote
3answers
58 views

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 ...
2
votes
2answers
47 views

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 ...
1
vote
1answer
49 views

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\}...
5
votes
1answer
123 views

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 $g$ ...
2
votes
2answers
113 views

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 $\sf{ZFO}$ ...
1
vote
1answer
28 views

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, ...
2
votes
1answer
114 views

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\{ \...
2
votes
2answers
23 views

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 ...
1
vote
2answers
98 views

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.
2
votes
1answer
37 views

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 ...
2
votes
1answer
49 views

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?
3
votes
1answer
34 views

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 $\...
3
votes
1answer
55 views

$X \times Y$ 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 ...
-1
votes
1answer
212 views

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 ...
0
votes
1answer
28 views

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 ...
1
vote
1answer
28 views

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 ...
0
votes
1answer
26 views

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 ...
1
vote
0answers
32 views

“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\...
4
votes
2answers
115 views

$f'(x)\geq 0$ 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 $...
0
votes
0answers
17 views

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 ...
-1
votes
2answers
19 views

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 ...
1
vote
0answers
55 views

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 ...
0
votes
0answers
41 views

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 ...
1
vote
1answer
50 views

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}(...
1
vote
0answers
28 views

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 ...
0
votes
0answers
28 views

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, anti-...
0
votes
0answers
18 views

Definitions Continuity Using the Order (Landau) symbolism.

Is an $o(1)$ function always continuous?
0
votes
2answers
35 views

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 ...
1
vote
2answers
33 views

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 ...
1
vote
1answer
52 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 =a\}...
1
vote
0answers
36 views

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 ...
0
votes
1answer
33 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?
2
votes
1answer
73 views

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, $T$, ...
0
votes
0answers
49 views

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 ...
4
votes
1answer
75 views

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 ...
1
vote
1answer
90 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 ...
3
votes
1answer
91 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 ...
1
vote
0answers
26 views

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} : \...
0
votes
1answer
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 ...
0
votes
1answer
47 views

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 ...
3
votes
2answers
89 views

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 ...
2
votes
0answers
52 views

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 $x,y\in\mathfrak{A}$)?
0
votes
0answers
31 views

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 ...
1
vote
0answers
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 $\neg(b<a)\...