0
votes
1answer
45 views

Why we use ANY in the definition of a maximal element?

I am confused about the following definition: "a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S." I do not understand ...
0
votes
2answers
55 views

How to define an isomorphism between $^\omega\omega$ and $\omega^\omega$?

Let $^\omega\omega$ be the set of all functions $x: \omega \to \omega$. Define $A = \{x \in ^\omega\omega \; | \; x \text{ has finite support}\}$, where by "finite support" I mean that the set $\{x(n) ...
1
vote
1answer
40 views

comparing two sets in set theory

I have two sets A and B and this condition holds: $\forall x \in A , y \in B: x \leq y$ Is there any standard term to describe the relation of A and B? something like $A \leq B$? Thanks for your ...
0
votes
1answer
64 views

Set theory, seems like a difficult question (Choice Function)

So, I don't know if this certain type of set has a name in English. If it has, I'll be glad if someone will edit my post. Definition: Let $A$ be a set and $f: P(A)\setminus \emptyset \rightarrow A$ ...
0
votes
1answer
22 views

Cardinality of partial orders on $\mathbb R \times \mathbb R$

Let the set $A$ be defined: $A=\{X \subseteq \mathbb R \times \mathbb R: X$ is a partial order $\land \max X \in \mathbb Z \land min X \in \mathbb Z \}$ What's the cardinality of $A$?
0
votes
1answer
23 views

If an infinite well-ordered set has initial segments of finite cardinality only, is the set isomorphic to $\mathbb N$?

Let $A$ be an infinite well-ordered set. Every initial segment of $A$ is finite. Is $A$ isomorphic to $\mathbb N$? What's the way to think about it? Should I build an explicit isomorphism? What ...
0
votes
2answers
24 views

Understanding first and minimal elements

Give an example of a set with partial order, that has a single minimal element but no first element. The example that was given is $\mathbb Z \cup \{2.5\}$, with relation $<$. I can't see why the ...
1
vote
3answers
26 views

Is (total-)order a weak version of countability?

One can easily see that countability also imposes or enables an ordering of the set. Now $\mathbb{R}$ is ordered but not countable. Is this ordering a weaker version of countability (at least in ...
1
vote
1answer
17 views

Antisymmetric relation (“strong” vs “weak”)

Defining: "weak antisymmetric relation": $\forall a, b \left< a,b \right> \in R \land \left< b,a \right> \in R \Rightarrow a=b$ "Strong antisymmetric relation": $\forall a, b \left< ...
1
vote
2answers
31 views

Order Type of $\mathbb Z_+ \times \{1,2\}$ and $\{1,2\} \times \mathbb Z_+$

I'm currently working on §10 of "Topology" by James R. Munkres. I've got a problem with task 3: Both $\{1,2\} \times \mathbb Z_+$ and $\mathbb Z_+ \times \{1,2\}$ are well-ordered in the ...
0
votes
2answers
28 views

Greatest lower and least upper bounds for a set of pairs

Had some trouble with this question in an exam recently, and wanted to make sure I reasoned correctly. The question was: $X$ is a set of pairs of real numbers $(x,y)$, with absolute values less than ...
0
votes
1answer
31 views

Showing a set is well-ordered

Let $(C,S)$ be a well-ordered set. Let $d \notin C$. We define the set $D=C \cup \{d\}$ and the relation $S'=S\cup (C \times \{d\})$. Show the set $(D,S')$ is well-ordered. Any help would be much ...
1
vote
1answer
26 views

Show that If $C$ is a chain in $X$ then $f(C)$ is also chain in $Y$.

Let X and Y are poset and $f:X\to Y$ is increasing function. If $C$ is a chain in $X$, show that $f(C)$ is also chain in $Y$. Since C is chain for every $x,y \in C: (x,y)\to \left(x\leq y\bigvee ...
0
votes
1answer
46 views

How to prove that $x\leq f(x)$ if f order isomorphic?

Let X is a well ordered set and $f:X\to f(X)=Y\subseteq X$ is order isomorphic. For any $x\in X$ prove that $$x\leq f(x)$$ since X is well ordered, {x,f(x)}$\subseteq$X and $x\leq f(x) $ or ...
1
vote
1answer
61 views

Prove/disprove questions on isomorphism and embedding between order types

About the notations: Let $\lambda, q, z, \omega$ be the order types of the reals, rationals, integers and natural numbers respectively. The sign $=$ means there's isomorphism and $\le$ means ...
4
votes
2answers
54 views

Is there a mistake in this question: $\forall a\in A: |\{x\in A:x\le a \}|=|\{ y\in B :y\le a \}|$?

Two ordered sets $(A,\le_A), (B,\le_B)$ and there's an isomorphic function $f:A\to B$ Prove $\forall a\in A: |\{x\in A:x\le a \}|=|\{ y\in B :y\le a \}|$ I think there's a mistake in this ...
1
vote
0answers
28 views

Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single ...
1
vote
1answer
41 views

Describing orders on sets $\{1,2,3,4\}$ and $\mathbb N$ with minimal/maximal/maximum/minimum elements

Describe all the partial orders on $\{1,2,3,4\}$ where the set of minimal elements are $\{2,4\}$ and the set of maximal elements is $\{1,3\}$ Describe all the partial orders on $\{1,2,3,4\}$ ...
2
votes
1answer
28 views

Continuity and Leximin

Consider a relation $\geq$ over the set of real-valued vectors. We say that $\geq$ is continuous if for any positive integer $n$, and any $\pi\in\mathbb Z^n,u\in\mathbb R ^n$ we have that the sets ...
4
votes
4answers
68 views

Number of join-irreducible elements of a lattice: is it monotonic?

Let $\mathcal L$ be a sub-lattice of $\mathcal P(X)$, where $X$ is a finite set. Denote by $\mathcal I(\mathcal L)$ the set of union-irriducible elements of $\mathcal L$ (i.e. $A\in \mathcal ...
0
votes
1answer
110 views

Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing singleton sets?

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
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
1answer
37 views

Something interesting about partial orders to show to my students

I'm looking for some interesting examples or theorems about partial orders to show to my students, can be also some fundamental theorem such as an analogue of this theorem on equivalence relations. ...
0
votes
0answers
23 views

Operation table of Hasse diagram

Consider the following Hasse diagram: My book gives the following join and meet operation tables for this diagram: $$\begin{array}{|c || c | c|} \hline Subset & x \wedge y & x \vee y \\ ...
1
vote
1answer
121 views

Exercise in Well Orderings

Prove that if $\prec$ is a well-ordering on a set X and if $Y \subseteq X$, then $\prec_Y=\{(x,y) \mid (x \in Y) \wedge (y \in Y) \wedge (x \prec y)\}$ is a well ordering on Y. I am a little ...
1
vote
0answers
24 views

Set theory chain proof help!

Let $A_i$ be a partially ordered set and $C_i$ a chain of $A_i$. Order $A_1\times A_2$ lexicographically. Prove that $C_1\times C_2$ is a chain of $A_1\times A_2$. This is a set theory proof ...
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
34 views

How to show a quasi-order ~ on a set S with relation // induces a partially ordered relation?

Let ~ be the quasi-order relation. and let // be defined as the relation s~t and t~s(s,t elements of S). Show that ~ induces a partially ordered relation on the set of equivalence classes relation //, ...
0
votes
1answer
22 views

Building an antichain in a finite poset

Given some finite poset $P$ we would like to find an antichain $A$ which intersects each maximal chain. How to do that? Note that each chain $C$ and each antichain $A$ intersects at one element as ...
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
22 views

Going down from filters to sets for specific relations

Let $\mathfrak{A}$ be a bounded lattice. I call a $2$-staroid a relation $f\in\mathscr{P}(\mathfrak{A}\times\mathfrak{A})$ such that $i\sqcup j \mathrel{f} b\Leftrightarrow i\mathrel{f} b\vee ...
3
votes
2answers
136 views

On Tarski-Knaster theorem

Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion. If $F: P(X) \to P(X)$ is monotone (order preserving), then $F$ has a fixed point. How ...
2
votes
2answers
325 views

Can it be proven/disproven that every set can be well-ordered to have a maximal element?

Can we prove or disprove that every set can be well-ordered to have a maximal element in ZF or ZFC? Or is it the area where different models have different answers to this?
0
votes
1answer
17 views

Diagonal Relation as a poset I - establishing the result by vacous truth

I have a problem with the logic behind the fact that, given a nonempty set X, the diagonal relation $D_X := \{ (x,x) : x \in X \}$ is a partial order on $X$. More specifically, my problem is with ...
1
vote
1answer
27 views

How to talk about overlapping partitions of a set?

If I have a set $A$ and a number of sets $A_\sigma$ for each $\sigma \in \Sigma$ such that $$A = \bigcup_{\sigma \in \Sigma}A_\sigma$$ Is there a concise and eloquent way of saying this in plain ...
4
votes
1answer
46 views

Problems with a proof that -in a linear order- a minimal element is the smallest element

I have a problem with a proof I found in Velleman's "How to prove it". This is sort of interesting, because it is the very first time I cannot see the structure of a proof presented in the book. The ...
1
vote
2answers
99 views

Initial Segments and Initial Sections of Posets

For a set A with a partially ordering <=, define the following 1) A subset s(x) of A = {y in A such that y <=x} 2) A subset S of A with the property that for every x in S then all y in A ...
2
votes
1answer
49 views

How can I prove that $(\mathbb Z,\leq_*)$ is isomorphic to $(\mathbb N,\leq )$

We define the binary relation $\leq_*$ between integers by the following rule: For $n,m\in\mathbb Z$, $n\leq_∗m$ holds if and only if one of the following conditions is satisfied: $|n|<|m|$, or ...
2
votes
3answers
58 views

I'm confused about the definition of poset.

The definition of poset : $\qquad$A set with a partial ordering. Partial ordering is a binary relation $\preceq$ over a set ($P$). If I understood the definition of relation correctly, then ...
-3
votes
1answer
180 views

What's the difference between relations, functions, and operations?

My understanding is vague: operations is a subclass of functions, functions is a subclass of relations. operations and functions can form terms but not formulas, relations can form formulas but not ...
3
votes
2answers
111 views

Why is the minimal uncountable well-ordered set $S_{\Omega}$ unique?

In section 10 of Topology by Munkres, the minimal uncountable well-ordered set $S_{\Omega}$ is introduced. Furthermore, it is remarked that, Note that $S_{\Omega}$ is an uncountable well-ordered ...
3
votes
1answer
32 views

Injectivity of rank function

Let $R$ be well founded and set like on $A$. I want to show that $R^{*}$ is a total order iff $rank_{A,R}(y)=\sup\{rank(x)+1|$ $xRy$ $\wedge{x\in{A}}\}$ is injective. Here $R^{*}$ is the transitive ...
2
votes
1answer
36 views

Likeness of strictly well-ordered sets

The question itself: Given a strictly ordered set $\langle A, < \rangle$ (< merely symbolises some kind of an order in this case) which is dense, with a least and greatest elemnts that are not ...
1
vote
0answers
34 views

The null set as the underlying set of a relation structure

Can the null-set underly a relation structure? Can it underly a well-order? If so, would such a well-order have order-type $0$? (Can one even have an order-type of $0$?) My motivation for this ...
0
votes
1answer
29 views

Question about sets of well-orderings

I recently posted a question on here about sets of well-orders and isomorphisms between well-orders. I have a related question about order-type-comparison: if $W$ is any set of well-orders, doesn't ...
5
votes
1answer
61 views

Question about sets of well-orders and isomorphisms between well-orders

I was thinking about this problem in trying to better get a feel for and understand well-orders (I'm trying to learn some set theory) - right now, I'm not really hoping for anybody to supply me with a ...
0
votes
1answer
34 views

Could someone check this proof for me?

The problem (and a definition first): The following definition explained Let $U \subseteq \mathbb{N}$. We say that a function $x: \{0,1\}^U\to \{0,1\}$ is a $xor$ on the set $U$ if the following ...
0
votes
2answers
32 views

How many pairwise nonisomorphic dense subsets exist

in the completion of a linear order of cardinality $\kappa$? Can there be more than $\kappa$? To be more precise, suppose $L$ is a linear order of cardinality $\kappa$. Let $\overline L$ denote ...
1
vote
2answers
104 views

Exercise in well-ordered sets

Definitions: Given a subset $A$ of a non-empty well-ordered set $X$ we define the supremum as follows: $\text{sup(A)}:=\text{min}\bigg(\big\{\,y\in X \oplus\{ +\infty\}: \forall x\in A \;(x\le y)\, ...
3
votes
2answers
79 views

The set of all the partial ordering of $X$ is a partially ordered set.

I have trouble with the following exercise. I don't understand it. Also I think that $x,y\in P$ should be $x,y\in X$. Could someone give an insight of the exercise and also some hint of how to solve ...