1
vote
0answers
26 views

Principal order filters on a POSET

I have another problem, but in this one I have no idea how to start. Let be $(X,\leq)$ a POSET with a first element and gifted with the topology $\{ (a,\rightarrow) : a \in X \}$ (principal order ...
2
votes
1answer
27 views

A problem concerning a partially ordered in $\omega$ and two chains.

Assume that $P=\left\langle\omega, \preceq\right\rangle$ is a partially ordered set such that for each $n \in \omega$ there are two chains $A_n$ an $B_n$ in $P$ such that $n \subset A_n \cup B_n$. ...
0
votes
3answers
57 views

Finite set with $\sup\{S\} \notin S $

Just a quick question about sets. Can anyone here think of an example of a finite set $S$ where the $\sup\{S\} \notin S $ ? I found this on a past exam paper for college, and am unsure if there ...
4
votes
0answers
36 views

Prove that for every 2 elements in the set F of all functions from N to N, there's an element in F that's bigger than both

let there be $\ F$ the set of all functions from $\ N \rightarrow N$. K is a relation on F, for every f,g$\in$F , (f,g)$\in$K $\leftrightarrow$ for all $\ n\in N$, $\ f(n)\leq g(n)$ Prove that for ...
0
votes
1answer
133 views

Show that a particular set is a poset

I would like to know if my understanding of the concept of a poset is correct. From what I've learnt from the class: A poset must be transitive, reflexive, and antisymmetric. Am I right? Therefore, ...
0
votes
1answer
41 views

Completeness of posets

I think this is rather a simple question in order theory, but if someone could explain it step by step that would be really useful. If we have an arbitrary set, then let's denote the set of all finite ...
0
votes
1answer
85 views

Is this poset a lattice?

Given the set $\Bbb Z^+\times\Bbb Z^+$ and the relation $$\begin{align*} (x_1,x_2)\,R\,(y_1,y_2)\iff &(x_1+x_2 < y_1 + y_2)\\ &\text{ OR }(x_1 + x_2 = y_1 + y_2\text{ AND }x_1 \le y_1)\;: ...
0
votes
1answer
77 views

Isomorphism between partially ordered sets - What is wrong with my argument?

I'm trying to show Isomorphism between two partially ordered sets is an equivalence relation. Suppose $M$ and $M^{\prime}$ are two partially ordered set and $f:M\to M^{\prime}$ is isomorphism ...
1
vote
1answer
59 views

Partially ordered set Question : $R$ relation on $\mathbb {N}$ iff $n\in \{1,5,5^{2},\dots,5^{k}\}$ , $x=n\cdot y$

I`m trying to prove that this relation is partially ordered set: $R$ relation on $\mathbb {N}$ iff $n\in \{1,5,5^{2},\dots,5^{k}\}$ and $x=n \cdot y$ the condition for partially ordered set are: ...
1
vote
1answer
53 views

number of antichains and number of downsets

This question is taken from the book Introduction to Lattices and Order by Davey and Priestley. Question 14 of Chapter 1. Definition : If $P$ is and ordered set and $Q\subseteq P$, then $Q$ is a ...
1
vote
2answers
133 views

Partially ordered set Question : $A=\{1,2,3,4,5,6\}$ ,$R =\mathcal P(A) \times \mathcal P(A) $

I`m trying to prove that this relation is partially ordered set: $A=\{1,2,3,4,5,6\}$ $R =\mathcal P(A) \times \mathcal P(A) $ $(B,C)R(D,E) \Longleftrightarrow (B \subset D) \vee ((B=D)\wedge(C ...
0
votes
1answer
35 views

Infinite linear ordered set which doesn't include infinte well-ordered subsets

Need an example of such set. Thank you for your time
0
votes
2answers
44 views

Composition of linear orders on the same set

I need to show an example of finite linear orders $K_1$ and $K_2$ are defined on the same set, such that their composition $K_1\circ K_2$ is not a transitive relation. Whatever orders I choose, their ...
0
votes
1answer
40 views

How to build linear order on $\mathcal P({\bf N})$ [duplicate]

How to build linear order on $\mathcal P({\bf N})$? Had idea about inclusion relation, but it does not satisfy linearity.
1
vote
1answer
44 views

strict ordering a Set

Given: $(A,<)$ is a strictly ordered set and $b \notin A$. Define: a relation $\prec$ in $ B = A \bigcup \{b\} $ as: $x\prec y $ if and only if $(x,y \in A \text{ and } x<y)$ or $(x \in A ...
2
votes
0answers
67 views

“Let A be a set. We are able to quotient all possible well-orders over the set A.” What does this mean?

"Let A be a set. We are able to quotient all possible well-orders over the set A." This was the first line in the set-up of some exercises I have to do (which ask specific questions depending on ...
3
votes
1answer
136 views

A problem about compact order topological space

This problem is from terrytao.wordpress.com/books/analysis-ii,Errata to the second edition (hardcover),P.390,Exercise 13.5.8 .I have difficulty proving this,appreciate any help! Show that there ...
2
votes
2answers
54 views

Finding the number of $2$-element chains of $B_n$

I'm trying to calculate the number of $2$-element chains and anti-chains of $B_n$, where $B_n$ is the boolean algebra partially ordered set of degree $n$. I understand that I want to take all of the ...
0
votes
1answer
138 views

Proving of existence of limit ordinal

How to prove that for every ordinal $\alpha$ there exist limit ordinal $\beta$ ,such that$\alpha\in\beta$ ? Thank you.
1
vote
0answers
60 views

Order relation with reversed order definition

I am trying to solve this exercise, but it is kind of confusing to me because the order relation involved "reverses" the standard order. Let $\tau$ a binary relation over $\mathbb N$ defined as ...
0
votes
2answers
353 views

Dedekind Cuts - Additive and Multiplicative Identities

I wanted to receive some help regarding some homework questions; I appreciate any sort of feedback possible. I am certain that any Dedekind Cut $A$ has the following properties: 1) $A$ is not the ...
2
votes
2answers
207 views

Well-Ordered Sets and Functions

I need some assistance with the following problems. I've come up with some ideas but may need some clarification. 1) Let $S$ be a well-ordered set. a) Need to show $S$ has a first element. ...
2
votes
2answers
169 views

isomorphism $\Bbb Q$ to $\Bbb Q \cap (0,1)$

I've got a question in my homework: Prove that $\langle \mathbb{Q},< \rangle$ and $\langle \mathbb{Q}\cap (0,1),< \rangle$ are isomorphic. I have tried to find a bijective function without any ...
0
votes
1answer
313 views

Maximal Elements - Ordered Sets

Problem 1: Let $S$ be an ordered set with a unique maximal element $x$. Again, I've worked some thoughts to these problems and would like to confirm their validity. I appreciate any feedback. (1) ...
5
votes
1answer
165 views

Exercise on partially ordered sets from Kunen's *Set Theory*

This problem is Exercise (F4) from Chapter VII of Kunen's text. Apparently, it is a result of Tarski. It goes as follows: Let $ \mathbb{P} $ be a partial order. Define $ \text{c.c.}(\mathbb{P}) $ ...
0
votes
3answers
94 views

$\forall{a\in A,\, b\in B} \mid a\le b$ Prove that: $\sup A \le \inf B$

Let A and B be two non-empty bounded subsets of $\mathbb{R}$. $\forall{a\in A,\, b\in B} \mid a\le b$ Prove that: $\sup A \le \inf B$ My solution goes as follows: Suppose $\sup A \gt \inf B$: ...
0
votes
1answer
259 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 ...
3
votes
1answer
170 views

How to derive distributivity from Boolean algebra laws

Let $(L,\le,\bot,\top)$ be a bounded lattice and $\neg: L \rightarrow L$ be a map that satisfies the following laws: $a \wedge b = \bot \Leftrightarrow a \le \neg b$ $\neg\neg a =a$ I'd like to ...
2
votes
1answer
170 views

Archimedean ordered fields and valuations (exercise)

Definitions: Let $(K,\leq)$ be a totally ordered field and $(G,\leq)$ a totally ordered abelian group (written additively). If we denote $\mathbb{Z}a\!=\!\{na;\, n\!\in\!\mathbb{Z}\}$, then $G$ is ...
1
vote
2answers
300 views

Draw Hasse diagram as two elements have same image

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$, $P=\{y \in \mathbb N : y \text { is a prime number}\}$, consider the map $f$ defined as follows: $$\begin{aligned} f:x\in S \rightarrow f(x) \in \wp (P) ...
1
vote
1answer
172 views

Uniqueness of meets and joins in posets

Exercise 1.2.8 (Part 2), p.8, from Categories for Types by Roy L. Crole. Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
3
votes
1answer
222 views

Examples of preorders in which meets and joins do not exist

Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
4
votes
2answers
346 views

partially ordered group, positive cone, quotient (exercise)

Definitions: A partially ordered group or po-group is a po-set $(G,\leq)$, such that $G$ is a group and $\forall x,y,a,b\!\in\!G\!:x\!\leq\!y\Rightarrow axb\!\leq\!ayb$, i.e. a po-set that is a group ...
4
votes
2answers
161 views

Stone duality for ideals and filters (exercise)

In A Course in Universal Algebra (Burris, Sankapannavar), the exercise 4.4.7-8, p.158, says: Let $A$ be a Boolean algebra. Denote $A^\ast:=\{\text{ultrafilters of }A\}$, and give $A^\ast$ the ...
1
vote
2answers
531 views

Prove König's theorem using Dilworth's theorem

I am trying to derive König's theorem from Dilworth's theorem, but it seems like I'm stuck. I know that I have to define some kind of binary relation on the set of a bipartite graph's vertices, then ...