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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Ask book to deeply understand partially ordered sets
I learn little about ordering and poset before, but I think it's not enough and want to learn more about ordering and Poset. Can anyone please recommend some best books to learn about this topic. I ...
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What's so well in having a least element in a set?
What's so well about the principle of well-ordering? Why is this pattern of order named like this? I am unable to trace a connection between well-ordering and a set that has a least element so I ...
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How to show that $(\Bbb Q, <)$ contains an order isomorphic copy of $\epsilon_0$ which is the least ordinal satisfies $\omega^\epsilon = \epsilon$?
We define $\epsilon_0 = \sup\{\omega, \omega^\omega,\omega^{\omega^\omega},\omega^{\omega^{\omega^\omega}},\ldots\}$. How to find a subset of $ \Bbb Q$, $(A, <_{\Bbb Q})$ that is isomorphic to ...
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Visualizing directed preorders
I am trying to understand directed preorders, a.k.a. directed sets. Are they analogous to connected DAGs?
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What do linearly ordered abelian groups look like?
Recently a post on MathOverflow connected a theorem about ordered abelian groups to the axiom of choice, and it made me want to try and play with these objects a bit.
But it appears to me that I ...
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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 ...
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“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 ...
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Looking for example of an order homomorphism that doesn't preserve joins.
I know that not every order homomorphism preserves joins. But, I can't think of an example!
Both minimal examples and 'natural' examples welcome.
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Why are the p-adic integers a linearly ordered group? [duplicate]
In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group.
I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
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Non-isomorphic posets
Is there any formula or counting algorithm for the number of non-isomorphic posets (defined on finite n-element set)?
I'm interested how to solve task *5, p.4 in Birkhoff's "Lattice Theory"
the book
...
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4answers
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Set that is not well-ordered
I want to find a set that is transitive, every non-empty subset of it has $\in$-maximum and is not $\in$-well ordered.
Any hint?
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Can all non-archimedean groups be written as a product of archimedean groups?
All the non-archimedean groups I know of can be written as the product of archimedean groups. I'm wondering if this is generally true. I think I've found a proof, but I haven't heard this theorem ...
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2answers
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Why does every countable limit ordinal have cofinality $\omega$?
According to Wikipedia, if $\alpha$ is a countable limit ordinal, then $\mathrm{cf}(\alpha)=\omega$. It is intuitively clear to me that it should be so. Certainly the cofinality of such an ordinal ...
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2answers
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Why Zorn's Lemma “fails” on nested boxes problem?
Backgound
My problem comes from the nested boxes problem. Consider a list of boxes. Each box has a length, width, and height. Since the boxes can be rotated those terms are interchangeable. The boxes ...
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Theory of structures with infinite Partial orders. $\langle H, \{\sqsubset_i \}_{i\in I} \rangle$
My recent interests have led me to have to deal with particular structures I have never seen before. Sets equipped with an infinite numbers of partial orders $\{\sqsubset_i:i\in I\}$.
I'm a bit ...
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Sufficient condition for equality with order hypothesis
I have this very simple equation for arbitrary $a,b,c,d,e,f\in\mathbb{R}$ with $\mathbb{R}$ endowed with the usual order :
$$a+b+c=d+e+f$$
So, if this equality is verified, I know that
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Enumerating all antichains in a finite poset
I have some reasonably small finite posets (on less than 20 points) and would like to iterate over all "downsets" in the poset, where a downset is a set closed under ≤ (so if x in X, and y ≤ x, then y ...
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Conditions for linearly ordered groups
One standard definition of a linearly ordered group is that the order $\leq$ obeys the law
(1) $a\leq b\implies ac\leq bc$.
Suppose (1) only holds for positive c. Must it also hold for negative ...
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a totally ordered set with small well ordered set has to be small?
doing something quite different the following question came to me:
1)If you have a totally ordered set A such that all the well ordered subset are at most countable, is it true that A has at most the ...
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Itō's Lemma neglecting terms
In my project I am trying to give a Heuristic proof of Itō's lemma. I show $E[dW_t^2] = dt$
I take $g(x,t)$ to be a twice continuously differentiable function and $dt$ to be infinitesimally small.
...
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Another way to state antisymmetry of a total order?
Adapted from Wikipedia:
"Set $S$ is totally ordered under $\le$" means:
$x,y \in S \implies ((x \le y \wedge y \le x) \implies x=y)$ (antisymmetry)
and
$x,y,z \in S \implies ((x \le y \wedge y \le ...
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Are there only 2 clopen sets on real plane?
How can I prove that the only open and closed sets on the real plane are empty set and real plane itself? Preferably by using order theory.
Thanks.
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Order-preserving isomorphism between $\mathbb{R}^n$ and $\mathbb{R}$
Suppose we have a linearly ordered group over $\mathbb Z^n$ where the ordering goes left-to-right, i.e. when deciding if $(x_1,x_2,\dots)<(y_1,y_2,\dots)$ we first check if $x_1< y_1$, if it is ...
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Generalization of metric that can induce ordering
I was wondering if there is some generalization of the concept metric to take positive and negative and zero values, such that it can induce an order on the metric space? If there already exists such ...
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Sub-(complete lattice)? - what's the correct terminology?
Let $X$ denote a set, let $\mathcal{O}$ denote the open sets of a topological space with carrier $X$, and let $\mathcal{P}$ denote the powerset of $X$. Furthermore, let $\leq_\mathcal{O}$ denote the ...
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$T\vDash\psi$ equivalences
$T\vDash\psi$ means $T$ satifies $\psi$ from Tarski's definition of truth, it simply means that the sentence $\psi$ is valid in $M$. I call a sentence $\psi $ universally if it is valid in every ...
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Semi-formal language - Universe has at least three elements
First of all I would like to construct a semi formal sentence, such that the universum has at least three elements. My attempt:
$$\exists x\exists y\exists z (x\not=y\wedge y\not=z\wedge x\not=z)$$
...
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How adding elements effects properties of posets
Given a poset $P = (X, \le)$, I am trying to create a new poset by adding a new element $x^*$ to $X$ and the ordered pair $(x^*,x^*)$ to the relation.
How does this effect the height, width, the set ...
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Name of the order with irreflexivity, antisymmetry and transitivity?
I have an order otherwise poset aka partial order but it is irreflexive so relationships such as 1R1 and 2R2 are impossible. What is the name of this order?
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A generalization of Galois connections
Let $f(x)\ast y \Leftrightarrow x\ast g(y)$ for some binary relation $\ast$ and functions $f$ and $g$. This is a generalization of Galois connections.
Are things like this studied before?
Note that ...
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A partial order with specified properties
I will say that two elements $a$ and $b$ of a poset are intersecting ($a\not\asymp b$) when there exists non-least element $c$ such that $c\le a$ and $c\le b$.
Let $\ge_0$ be a partial order on a set ...
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Example of a Partial Order which is not a Total Order and Why?
I'm looking for a simple example of a partial order which is not a total order so that I can grasp the concept and the difference between the two. An explanation of why the example is a partial order ...
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Infinite set ordered like a “infinite tree”
Obviously I don't need to say that I'm still a newbie, so I'll try to explain my question with some pics too.
I have a infinite set $T$ I have a visual idea of how this set is ordered, but I don't ...
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Exclude operation symbols in signature
We probably know what a signature is, it contains a set $\sigma_{op}$ (the operation symbols), $\sigma_{rel}$ (the relation symbols) and a function $ar:\sigma_{op}\cup\sigma_{rel}\rightarrow\mathbb ...
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Semi-formal language
I want use semi-formal language to describe the following four points.
(1) The group axioms with signature $\{*\}$
(2) The property "linear order" with signature $\{<\}$
The following properties ...
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Strict partial order
I know what is a partial order: for example the power set of a set or the natural numbers.
But a strict partial order is a set with a binary relation $R$ so that $R$ is transitive, irreflexive (not ...
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“Lexicographic order” without priority, but with ties, how to define / what's the name?
I'm searching for an ordering principle which combines two (or more) preorders, akin to the lexicographic order, but not quite: I'm looking for an ordering principle that does not priorities, but ...
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Dense linear orderings isomorphism [duplicate]
First of all the definition: A dense linear order is a $\{<\}$-structure in which the following formulas are valid:
Question: How can I prove that any two countable DLO's (dense linear order) ...
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Projecting external points to a circle: Distance order preserving?
Given a circle, and a set of points $A$ that lie external to the circle; I perform the following simple operation:
I compute the point of intersection of the i) circle and the ii) line joining each ...
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Finding an ordered set
Find a partially ordered set $A$ so that for every member a in $A$, there is $b∈A$ such that $b<a$ but for infinitely many members of $A$, there does not exist $c\in A$ such that $a<c$.
Thanks.
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Finding a partially ordered set
Find a partially ordered set $A$ so that for every member $a$ in $A$, there is $b\in A$ such that $b<a$ and there exist just finitely many members bigger than $a$.
Thanks.
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2answers
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Partially ordered set, maximal and minimal
Show there exist partially ordered sers with more than one maximal element and /or more than one minimal element. This is not a Homework question so completeness is appreciated.
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Subset of a partially ordered set: Least upper bound and greatest lower bound.
Let $(A, \leq)$ be a poset and $B \subseteq A$.
I need to show
i) B may have at most one least upper bound in A
ii) B may have at most one greatest lower bound in A.
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order preserving function over Poset
I have a DAG (Directed Acyclic Graph) $G$ represents the dominance relation between solutions. A path from $x$ to $y$ means $y\succ x$ ($y$ dominates or better than $x$) where the absence of such a ...
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Do the pth powers of $p$-norms define the same partial ordering on the set of all probability distributions for all $p>1$?
Consider the $p$-th power of the Schatten $p$-norm $||q||_p$ of a probability distribution $q$ , ie, the function $\sum_j q_j^p$, where $\sum_j q_j = 1$ and $q_j \geq 0$. For fixed $q$ and $p>1$ ...
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Help on total ordering and partial ordering
Hi guys my final exam is coming up, and I am given a few practice problems and I don't get how to do these questions on total ordering and partial ordering. Here are the question and here is how I did ...
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Give an example of a simply ordered set without the least upper bound property.
In Theorem 27.1 in Topology by Munkres, he states "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact."
(The LUB ...
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Is lexicographic order complete?
I was wondering, if the lexicographic order in $\mathbb{R}^2$ is complete or not? I guess it isn't but I dont find any counterexample.
Complete means: For every partition of $M$ into two disjoint ...
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Not sure about this Hasse Diagram
I'm not sure how to draw Hasse diagram of this set $\{-2,-1,0,1,2\}$, $R = \{(a,b)\mid a\leq b\}$. My attempt is just a straight edge with the smallest vertex at the bottom and the greatest at the top ...
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Topological Sorting in Linear Order for Hasse Diagram
I have come across an exam review question that I am stuck on. The question states:
Use topological sort to compute a valid linear order of the elements for the following Hasse Diagram:
This is ...
