-1
votes
0answers
16 views

prove the statements of well ordered sets… [on hold]

Show that: a. If $(X,<)$ is a chain and finite set, then is it a Well Ordered Set? b. If $(X,<)$ is Well Ordered then any Y $\subseteq$ X is Well Ordered. c. If $(X,<)$ is Well Ordered and ...
0
votes
1answer
77 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
0answers
7 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 ...
1
vote
2answers
121 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
312 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
12 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
19 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
39 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
89 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
46 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 ...
0
votes
0answers
17 views

Lattice of reduced covers

What are reduced covers for a set. How to draw a lattice with the reduced covers. Please explain with an example for a set of size 3. How to count number of reduced covers of a set.
0
votes
0answers
7 views

Difference between LUB of reduced covers and partitions.

Given a set, how is LUB different in case of reduced covers of the set and partitions of the set. Please explain with an example.
2
votes
3answers
57 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 ...
-4
votes
1answer
101 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
51 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
31 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
34 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
33 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
28 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
56 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
31 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
31 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
83 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
74 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 ...
4
votes
2answers
37 views

disconnected ordered set

Is there a totaly ordered infinite set $A$ with the least element $a$ and the greatest element $b$ such that for any sequence $\{\alpha_n\}$ and $\{\beta_n\}$ in $A$ which satisfies ...
6
votes
1answer
187 views

Intuition behind proof and verification partially ordered sets

Hi everyone in the book that I read I have trouble to understand the argument of the proof at the below proposition. There is a lot of point which are left as exercises, which is great. One of these ...
2
votes
1answer
105 views

Exercise: Every non-empty subset of $X$ contains a minimum and maximum element and the converse

Hi everyone is my second exercise of posets. And find the following, the first part I think is not difficult at all. But for the converse I have some serious troubles to prove it. I have two ...
2
votes
0answers
43 views

Set Theory - Well Order (Lexiographical combination)

Question: Prove constructively that if $(A_{1},\prec_{1})$ and $(A_{2},\prec_{2})$ are two well-ordered sets then their lexicographical combination $(A_{1} \times A_{2},<_{1,2})$ is also well ...
1
vote
1answer
58 views

Are there any known uncountable transfinite increasing sequences of real numbers?

The real numbers are uncountable, so assuming the axiom of choice there is at least transfinite sequence of real numbers $r_0, r_1, r_2, ..., r_\omega, r_{\omega + 1}, ..., $ up to (and possibly ...
0
votes
1answer
43 views

Could the empty set be complete partial order?

According to the definition, $(S,\leq)$ is cpo when each totally ordered subset of $S$ has supremum. I'm sure $(\emptyset,\emptyset)$ is a total order, but I don't know whether $(\emptyset,\emptyset)$ ...
5
votes
2answers
62 views

Countability of a certain set

Suppose $S$ is a totally ordered set such that for any $x,y\in S$ there are finitely many elements between them. Does it follow that $S$ is (at most) countable? It seems like the answer should be ...
1
vote
1answer
48 views

How many complete theory extend theory T are there?

$Т$ - theory of signature $\{\le\}$, defined by the axioms А1-А5: А1: $\forall x(x=x)$ А2: $\forall x,y\big((x\le y\land y\le x)\to x=y\big)$ A3: $\forall x,y,z\big((x\le y\land y\le z)\to x\le ...
1
vote
1answer
50 views

Proof that there is an order relation

For an arbitrary set M there is a relation $R \subseteq 2^M \times 2^M$ about $$ A \mathrel R B \Leftrightarrow A \cup \{x\} = B$$ The join is a disjoint join. There are not more details what is $x$. ...
2
votes
3answers
53 views

Non-trivial order on a set with infinitely many minimals/maximals

I am looking for an example of each: 1) A non-trivial order on a set A such that there are infinitely many minimal elements and 2) A non-trivial order on a set A such that there are infinitely many ...
2
votes
1answer
44 views

How to rigorously prove that these two sets have different order types?

Let $A$ and $B$ be two given ordered sets with the linear (or total) order relations $<_A$ and $<_B$, respectively. Then $(A,<_A)$ and $(B,<_B)$ are said to be of the same type if there ...
1
vote
1answer
51 views

I want to show that $\mathrel{R_1}$ and $\mathrel{R_2}$ are a partial order on $\mathbb N$, how would I do this?

Let $\mathrel{R_1}$ and $\mathrel{R_2}$ be relations on $\mathbb N$ defined by $x\mathrel{R_1}y$ if and only if $y = a + x$ for some $a \in\mathbb N_0$. $x\mathrel{R_2}y$ if and only if ...
1
vote
1answer
29 views

Is this complete partial order?

Is $(\mathbb{N} , \#)$ complete partial order, where $m\#n$ iff $(\exists k \in \mathbb{N})m=kn$. I proved it's partial order. For completeness I take directed subset and I know there is upper bond ...
0
votes
1answer
87 views

How to tell if relation on set is a partial order when relation is defined as a set of ordered pairs?

Determine whether $R$ is a partial order on set the $S$ and justify your answer. $S=\{1,2,3\}$ and $R=\{(1,1),(2,3),(1,3)\}$. So the job is to consider if $R$ is reflexive, antisymmetric and ...
4
votes
1answer
70 views

order preserving implies isomorphism

Let $M = (M, <)$ and $N = (N, <)$ be two dense totally ordered sets without endpoints. Let $F$ be the collection of order preserving maps between finite subsets of $M,N$ respectively. ...
2
votes
1answer
104 views

If $A$ is well ordered and $B$ is well ordered then $A\times B$ with the lexicographic order is well ordered

I am trying to prove that: Given that $A,B$ are well ordered sets, also $A \times B$ is a well ordered set. (A set $A$ is well ordered if a linear order is defined on it, and every non empty subset ...
0
votes
3answers
82 views

How to show that $ \succ acyclic \implies \succ asymmetric $

For a preference relation defined as $$ \succ := \{ (x,y) \in X\times X : x\ is\ better\ than\ y \}$$ one has to show that $$ \succ acyclic \implies \succ asymmetric $$ whereas $$ acyclic := ...
1
vote
1answer
74 views

Every countable lattice has a cofinal totally ordered subset?

If a lattice is countable, prove that it has a subset that is both totally ordered and cofinal in the lattice. Cofinal means that for each $l$ in the lattice, there is some $a$ in the subset such that ...
0
votes
1answer
179 views

Non-isomorphic countable linear orders which embed into each other

Def: $(A,<) \hookrightarrow (B,\lhd)$ if $\exists f:A \to B$ such that $\forall a,b \in A[a<b \implies f(a) \lhd f(b)]$. May I seek your advice on how to construct two infinite, countable ...
0
votes
1answer
56 views

Prove that $R$ is a linear order on $A\times B$

Assume that $(A,\leq)$ and $(B,\leq)$ are linearly ordered sets. Define a relation $R$ on $A\times B$ by $(a,b)R(c,d)$ iff $a<c$ or both $a=c$ and $b\leq d$. Prove that $R$ is a linear order on ...
3
votes
2answers
57 views

Why does sup-dense imply inf-continuous?

I am confused. The question is as follows: Let $(S, \preceq$), be a sup-dense subset of $(T, \preceq)$, both partially ordered sets. In other words, for every $t \in T$, there exists a subset $E ...
3
votes
1answer
61 views

Given $A \subseteq X$ we can have $\sup A < \inf A$?

For some poset $(X,<)$ with a global max $\top$ and min $\bot$ with $\bot < \top$, and a subset $A = \varnothing$, we'd have $\sup \varnothing = \bot$ and $\inf \varnothing = \top$, right? So ...
0
votes
2answers
71 views

What is an order reversing bijection on $\Bbb R$?

What is an order reversing bijection on $\Bbb R$ and why must it be continuous? Thanks
2
votes
2answers
122 views

Universality of $(\mathbb Q,<)$

I have few questions to the proof on Universality of linearly ordered sets. Could anyone advise please? Thank you. Lemma: Suppose $(A,<_{1})$ is a linearly ordered set and $(B,<_{2})$ is a ...
2
votes
1answer
62 views

Equality and order in sets

Just started Baby Rudin and got struck in this. While defining order in sets, $<$ was introduced as a relation and for a set to be ordered the condition was: for all $x,y$ belonging to an ordered ...
1
vote
1answer
106 views

Order-reversing bijection on partially ordered sets

Let $(X, \preceq)$ and $(Y, \ll)$ be partially ordered sets. Suppose there are functions $f : X \to Y$ and $g : Y \to X$ such that $$x \preceq g(y) \iff y \ll f(x)$$ for all $x \in X$, $y \in Y$. ...