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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Directed sets to describe a topology with nets.

I'm studying some things related to ultrafilters on metric and topological spaces and trying to prove theorem in a general setting, so the following question came to my mind. Let $S$ be a ...
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How to formalize a lattice in a graph

Given directed a graph: G = (V, E). How to use algebra symbols to express a lattice in G? where reachability stands for partial order, i.e. in the lattice ...
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Gallian: is it true that the well-ordering principle can't be proven from properties of arithmetic?

I just started reading Gallian's Abstract algebra and on page 3 it says "An important property of the integers...is the so-called Well Ordering Principle. Since this property cannot be proved from the ...
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Hasse diagram question about relations

I have the following Hasse diagram below, the question is given specific generalised quantifiers I have to list the subsets of {a,b,c,d} which the quantifier corresponds to. I have completed the ...
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3answers
125 views

Munkres Section 10: How are these order types different?

Let $\mathbb{Z}^+$ denote the set of all positive integers in the usual order, let $n$ be a positve integer, and let the following sets have the dictionary order: $\{1, \ldots, n \} \times ...
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+50

Generalized semilattice morphism

Join-semilattice morphism from a join-semilattice $\mathfrak{A}$ to a join-semilattice $\mathfrak{B}$ is a function $\alpha$ conforming to the formula $\alpha(X\sqcup Y) = \alpha X\sqcup\alpha Y$ ...
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1answer
37 views

Difficulty understanding “antichains”

I'm having a difficulty understanding the concept of "antichains". An antichain in a poset $P$ is a subset $C \subseteq P$ such that no 2 elements are comparable I understand the definition, ...
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9 views

Can we distinguish direct from transitive partial orders

If I have a set L and a partial order ≤ defined on L, and (L, ≤) is a lattice. We know that ≤ is transitive, i.e. if l1 ≤ l2 and l2 ≤ l3 then l1 ≤ l3. Well. I ...
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29 views

How to prove that if R is a partial order then also $R^{-1}$ is also a partial order?

My guess was to divide the problem into 3 cases: prove that $R^{-1}$ is reflexive, antisymmetric and transitive. To prove that $R^{-1}$ is reflexive, I did: If $R$ is reflexive, that follows that ...
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28 views

Does this combinatorial object have a name?

I have come across the need to with subsets of meet-semilattices. Specifically, my setting is that I need posets that have a meet operation that is unique when defined, but is not necessarily defined ...
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38 views

Prove the relation on $\Bbb N \setminus \{0,1\}$ is a partial order

I'm a bit new to this material and trying understand some problem I'm solving Let $R$ be a relation on the set set = $ \{ N \setminus \{ 0,1\}\} $ that's defined like this: $aRb$ if there is an ...
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25 views

Are the face posets of CW-complexes Eulerian?

Suppose we had a CW-complex $X$ with decomposition $X_{i}$ Is its face poset, consisting of cells and covers generated by the attachment of cells to one another, an Eulerian poset? What would be the ...
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61 views

Are the Hyperreals complete?

Since $^*\mathbb{R}$ does not form a metric space then it can not satisfy the Cauchy conditions for completeness. However, my intuition is telling me that it would satisfy conditions of completeness ...
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1answer
29 views

Proving distributivity of Heyting algebras with the Yoneda lemma.

How can one prove distributivity of a Heyting Algebra via the Yoneda lemma? I'm able to prove it using the Heyting algebra property $(x \wedge a) \leq b$ if and only if $x \leq (a \Rightarrow b)$. ...
0
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1answer
20 views

for any subset S of finite group G and any g,h∈G, show |gSh|=|S|

Let $G$ be a finite group. For any subset $S$ of $G$, and any group elements $g,h \in G$, I want to show that $|gSh|=|S|$, i.e. that the cardinality of $S$ is preserved when multiplying on the left ...
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1answer
48 views

Question about supremum $\implies$ infimum

For all the proofs I've seen that if a set has the least upper bound property, then that set also has the greatest lower bound property, they assert something like if a set has lower bounds, then the ...
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1answer
47 views

Is relation a partial order?

can you give me few hints how to solve this problem ? Relation R on the set P(A) A = {a,b,c,d} is a set of four elements. We also have relation R on the set P(A), which is defined R={(A,B)│A ⊆ B. ...
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1answer
38 views

How can find this matrix?

Let $U$ be an $k\times n$ matrix and $G$ an $n \times n$ matrix over $\mathbb F_q$. We know that valuation of $UG$,$UG^2$,...,$UG^{k-1}$. ($k$ is the order of $G$) If we dont know components of $U$ ...
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1answer
45 views

Möbius inversion for locally finite poset

I'm trying to understand this paper: Rota, Gian-Carlo. "On the foundations of combinatorial theory I. Theory of Möbius functions." Or at least, to understand the first few pages. In particular, I'm ...
0
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1answer
16 views

Minimal and maximal element in a subset of a Kolmogorov space.

Let $X$ be a Kolmogorov space and define on it a specialization order, ie.: for any $x,y \in X$, we have $x \leq y$ if and only if $x \in \overline{ \{y\} }$, where $\overline{ \{y\} }$ is the ...
10
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3answers
506 views

Sole minimal element: Why not also the minimum?

A minimal element (any number thereof) of a partially ordered set $S$ is an element that is not greater than any other element in $S$. The minimum (at most one) of a partially ordered set $S$ is an ...
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2answers
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Difficulty understanding Zorn's Lemma.

This is a question on chap. 9 of Introduction to Set Theory of Hrbacek and Jech. The book define that the Zorn's Lemma is: If every chain in a partially ordered set has an upper bound, then the ...
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1answer
71 views

Another way to express certain filter

Let $\mathfrak{A}$, $\mathfrak{B}$ be posets, $\mathfrak{A} \subseteq \mathfrak{B}$ (and $\mathfrak{A}$ is the induced order of $\mathfrak{B}$). Let $\mathcal{A}$ be a filter on $\mathfrak{A}$. Help ...
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1answer
139 views

Does the ordered set from Zorn's lemma have a lowest element?

The ordered set in the Zorn's lemma needs to have supremums for all chains, that is, including the empty chain. The supremum of an empty chain is its lowest upper bound. Since any element is an upper ...
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69 views

an example of the Scott topology

Let $X=A\cup B$ where $A$ is set of all positive integers w.r.t the Scott topology and $B$ is singleton set $\{b\}$. A set $U$ is open in $X$ iff $U$ is open in $A$ or $A\subseteq U$ and $U\cap B ...
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21 views

Types of ordering

Can somebody please help me understand how ordering of numbers work? There are 3 types of ordering I want to understand. Lexicographic, reverse lexicographic and Fike's ordering. How would the ...
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1answer
26 views

I can not understand some steps in Tverberg's proof on Dilworth theorem

As is shown in the underlined part of the picture above. First, why after $c^+ \not\in S^-$, we can use induction hypothesis? how does it imply $S^-$ contains no antichain of cardinal $m + 1$? ...
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35 views

Orders, inf & sup of sequence

So I'm completely new to the idea of isomorphism and orders and I have to show, if the following sequence is bounded and find its inf / sup or show that it doesn't exist. $[{(0,1)^*\cup ...
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1answer
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1answer
19 views

How would i apply Dilworths Theorem to the following set: $S=\{ 0,1,4,6,7,8,9\}$ where any element $a$ is less than or equal to $b$?

This is the set $S=\{ 0,1,4,6,7,8,9\}$ under the order defined by divisibility. I know we have to find antichains and chains, and that the maximum number of partitions of $S$ into chains should equal ...
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1answer
16 views

Linear order where every initial segment is finite

It's true that a set with a linear order where every initial segment is finite is necessarily countable? I don't think so but I can't find a counterexample (or make a proof)
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Is there a linear order with this property

I was trying and failing to construct a linear order L each of whose uncountable subsets contains an uncountable well ordered subset but L is not a countable union of well ordered subsets. Is this ...
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131 views

Is there a name for those relations that behave a bit like $<$?

Consider a fixed but arbitrary preordered set $X$. Is there a name for those binary relations $R$ on $X$ satisfying the following? They seem to show up a lot. (Note that every such $R$ is necessarily ...
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73 views

An isomorphic map from natural numbers to positive rational numbers that preserves addition, multiplication and order [on hold]

Since $\mathbb{Q}^{+}$ is countable, there is a bijection between $\mathbb{Q}^{+}$ and $\mathbb{N}$ (0 included). Then the question now is, can we go further by constructing an isomorphic map between ...
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Embedding $\omega_1$ in the hyperreals

I've been trying to show that there exists a subset of the hyperreals which has order type $\omega_1$, that is, that there exists a subset $A \subseteq {}^*\mathbb{R}$ for which there is an order ...
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1answer
21 views

Showing certaing Integral domain is not well ordered.

Let $\mathbb{Z}[\sqrt2]=\{ a+b\sqrt2 ~| a,b\in \mathbb{Z}\} $ be an integral domain. Let $p=\{ a+b\sqrt2 ~|a,b\in\mathbb{Z} ~~~ and ~~~ a+b\sqrt 2 ~~is~~a~positive~real~number \} $ Show that ...
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1answer
78 views

Why is this not a category?

Why need the composite of two monotone functions not be monotone? This is from Rings and Categories of Modules, Anderson., Fuller., page 7
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41 views

What are the orders of modulo 9 and modulo 28? [closed]

I know what is the order, but unfortunately I can't calculate it. Please help me if you can, thank you very much.
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7 views

Form Poset from (Non-zeta) Matrix

I have a matrix (well, really a table) of values that are the results of various indicator systems. Naturally, each system is slightly discordant, so no one true ranking exists. I'm looking to form ...
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1answer
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Chain of elements of a chain of order ideals

Let $P$ be a partially ordered set, and let $I(P)$ be the set of order ideals (i.e. downward-closed sets) in $P$ ordered by inclusion. Suppose that $X$ is a chain of non-empty elements of $I(P)$. ...
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Dedekind(?) representation lemma on posets?

Here's an easy lemma: Any poset $(S, \preceq)$ is order-isomorphic to a subset of the powerset $\mathcal{P}(S)$ ordered by set-inclusion. I seem to recall having seen this attributed to Dedekind. ...
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1answer
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Is there an orderpreserving bijection from $\mathbb R$ to $\mathbb R-\{0\}$?

Looking at $(\mathbb R,<)$ where $<$ denotes the usual order I understand that $\mathbb R-\{0\}$ will inherit an order (as every subset of $\mathbb R$). I could not find essential differences ...
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1answer
30 views

When can $\sup_x$ and $\sup_y$ be swapped?

$f$ is a mapping from $X \times Y$ to $Z$. $X$ and $Y$ are sets. $Z$ is a set with $\sup$ defined ( I am not very sure how to formulate that. Is $Z$ required to have some total order?), and can be ...
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1answer
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A question about cofinal and coinitial order-preserving injections between linearly ordered sets

There is an inclusion $f : \mathbb{Z} \hookrightarrow \mathbb{R}$. Furthermore, if $A \subseteq \mathbb{R}$ has a maximum element, then so too does $f^{-1}(A)$. There is also an inclusion $g : ...
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1answer
32 views

Lower Bound for subsets of $\mathbb{N}$ [duplicate]

Let $P = \{A \in \mathbb{N} : A \neq \emptyset, |A|<\infty, |A| = 2k, k\in \mathbb{Z}^{+}\}$. Consider $X = \{ A \in P: 1 \in P \}$. Does $X$ have a lower bound in $(P,\subseteq)$. We can see that ...
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1answer
36 views

Does X have a lower bound?

Let P={A $\subseteq$ $\mathbb N$: A is non-empty, finite and has an even number of elements}. Consider X={A $\in$ P : 1 $\in$ A}. Does X have a lower bound in (P,$\subseteq$)? I think that there is a ...
2
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1answer
36 views

For any $a\in S_\Omega$, either it has an immediate predecessor or there is an increasing sequence in $S_\Omega$ whose l.u.b. is $a$. [duplicate]

Specifically I would like to show: Let $S_\Omega$ be the minimal uncountable well-ordered set. For any $a\in S_\Omega$, either $a$ has an immediate predecessor in $S_\Omega$, or there exists an ...
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1answer
8 views

does isomorphic Hasse diagrams represent two different posets or same poset?

I want to have a clear picture of isomorphic hasse diagrams, what do they represent, same poset or two different posets?
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Why is every monomial order a well-ordering?

Let $R$ be a commutative, unital ring. According to wikipedia, a monomial order is a total order $\leq$ on the space of monomials of a given polynomial ring of $R$ such that If $u\leq v$ and $w$ is ...
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Prove that, for all ordered sets P, Q and R,

$\langle P \rightarrow \langle Q \rightarrow R\rangle \rangle \cong \langle P \times Q \rightarrow R\rangle $ where $\langle Q \rightarrow R\rangle $ is the set of all order-preserving maps from Q ...