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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Is there a concept of |x|<0?

I'm just curious, did any of the famous mathematicians consider |x|<0? Would the zero have to be then not the additive identity and what else would it be then? (assuming you could build a field ...
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348 views

How many partial derivatives does a multivariate polynomial have?

My motivation for this question is from the following toy example; define the (nondeterministic) finite automata generated by the nonconstant* polynomial $f(x_0 , \dots , x_n) \in \mathbb{Z} [x_0 , ...
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Can someone explain this: “the set of subspaces of a vector space ordered by inclusion”

This is a claim on Wikipedia https://en.wikipedia.org/wiki/Partially_ordered_set I am not sure how to make sense of the claim What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? ...
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Find an example such that $X$ with the lexicographic order is not well-ordered.

Let $\{A_n\}_{n\in\Bbb N}$ be a collection of well-ordered sets. $X$ is defined by $X=\prod_{n\in\Bbb N}A_n$. Find an example such that $X$ with the lexicographic order is not well-ordered. I know ...
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Are $(\mathbb R\times \mathbb Q, \le_h)$ and $(\mathbb Q\times \mathbb R, \le_h)$ isomorphic? [on hold]

Are $(\mathbb R\times \mathbb Q,\leq_h)$ and $(\mathbb Q\times \mathbb R, \leq_h)$ isomorphic? when "$\leq_h$" is the right lexicographic order? Thanks a lot!
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75 views

What does it mean to say “a divides b”

I am not a number theorist and I am learning about relations. I encountered a relation that says $a \leq b$ if $a$ divides $b$ Can someone clarify what it means to a number to divide another ...
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647 views

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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25 views

(Order Relation) Monotone Continuous Complete Preorder on $\mathbb{R^L_+}$ has $y\geq x\rightarrow y\succsim x$

I am trying to show a monotone continuous complete preorder on $\mathbb{R^L_+}$ has $y\geq x\rightarrow y\succsim x$. Can you please share your 2cent on the below proof? Thank you! Point of ...
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87 views

Preference relations that are order-separable

This question concerns dense order relations and separability of orders. This question has changed radically since the first version. Hence the first two answers below now look strange because they ...
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49 views

A well-order on a uncountable set

I can't find an example of a well-order on an uncountable set. Is possible to prove that exists with the Axiom of Choice? How can I give a pratical construction? I try to define a well-order on ...
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Determine whether or not the poset $(\mathbb{N}, \propto$) has a least element

"Define a relation $\propto$ on the natural numbers $\mathbb{N}$ by declaring that for $x, y \in \mathbb{N}$, $x \propto y \iff (x=y)$ or $(3x \leq y)$ a) Show that $\propto$ is a partial order on $\...
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41 views

Order Theory and Lattice Theory Synonymous?

Is Order Theory the same as Lattice Theory? Can anyone recommend good beginners text book on either?
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52 views

Greatest Galois connection

Let $\mathfrak{A}$, $\mathfrak{B}$ be bounded posets. The main question: Explicitly describe (and prove that it exists) the greatest Galois connection between $\mathfrak{A}$ and $\mathfrak{B}$ (as ...
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22 views

Infimum and supremum of subset of inclusion Power set

I'm having trouble understanding the following exercise: Given $U=\{1,2,3,4\}$ with $A=P(U)$ the power set of the elements of $U$ and $R$ the inclusion relation over $A$. Determine the infimum and ...
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14 views

Looking at direction in complex numbers and vector analysis as a cartesian products of 2 imaginary and real total orders.

Can we abstract the idea of direction into an a cartesian product of two total orders? for example:$T_R = \{a,b,...\}$ and $T_I =\{ai,bi,...\}$ where ${T_R}×{T_I}$ is all possible directions and the ...
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Graph properties of Bruhat order for the general linear Lie algebra $\mathfrak{gl}$ on $\mathbb{Z}^n$

Let $P = \oplus_{i\in \mathbb{Z}}\mathbb{Z}\epsilon_i$ the free abelian group of infinite rank. Then we have a natural partial order $\leq'$ on $P$, that is, $a \leq' b $ if and only if $b \in a+\sum_{...
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How many ways can all six numbers in the set $\{4, 3, 2, 12, 1, 6\}$ be ordered

Is there an easy way to solve the problem? How many ways can all six numbers in the set $S = \{4, 3, 2, 12, 1, 6\}$ be ordered so that $a$ comes before $b$ whenever $a$ is a divisor of $b$? By ...
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1answer
38 views

Can a partially ordered set always be partitioned by its chains?

I would like any references that discuss how to partition a partially ordered set into subsets such that each subset is a chain in the partial order. For example, I am thinking that any partial order (...
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1answer
57 views

Can someone reconcile the two definition of Suslin's condition?

I am given two definitions of the so called "Suslin's condition" and I need to reconcile them. I am an undergrad and this is just for exploration. Definition 1. A partially ordered set $X$ is said ...
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39 views

Why is Boolean a lattice?

I've had minimal exposure to lattice theory but I must answer this question due to a project I'm working in. If anyone could answer this question in the simplest explanation possible with examples ...
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14 views

How to turn a NON-strict total order into a strict total order with $R^3$ vectors?

I'm currently working with colored images in the RGB color space. It's trivial to find a ordering in grayscale images (each shade of gray can be though as a value and darker shades comes before than ...
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32 views

Closure of Poset $Q_n = \{x : x \mid n\}$

Let $(S, <)$ be a poset. A smallest poset $(S', <)$ is called a closure of poset $(S,<)$ iff $S$ is a subset of $S'$, $\operatorname{glb}(x,y)$ is in $S'$, and $\operatorname{lub}(x,y)$ is in ...
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There exists a partial ordering of $ \mathbb{R} $ with no uncountable chains or antichains

here's an exercise I'm stuck upon. I'm not sure how to approach this: There exists a partial ordering of $ \mathbb{R} $ with no uncountable chains or antichains While looking for some help, I ...
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48 views

Functions from $ \kappa $ to $ 2 $ ordered lexicographically

I'm wondering how to approach the following exercise: Suppose $ \kappa \geq \omega $. Consider the set of all functions $ \kappa \rightarrow 2 = \{0,1\} $ ordered lexicographically. Show that ...
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Decimal representations of the reals both 1-1 and order-preserving?

The conventional representation of the non-negative reals as infinite decimals is 1-1 for numbers which are not multiples of $10^{-n}$ for some $n$, and 2-1 for numbers which are (for example, $0.2999....
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Integer Partition Refinement in Sage

A partition of an integer $n$ is a non-decreasing list of positive integers summing to $n$. For example, $3$ can be partitioned as $1 + 1 + 1$, $1 + 2$ or just $3$, but $2 + 1$ is indistinct from $1 + ...
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Do the Arithmetic Operations form a Heyting Algebra?

I was reading an article about partial orders, and it gave an exercise to find a relation where multiplication formed meets and addition formed joins (it did not specify an underlying set, but it ...
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39 views

Finite generating sets and finitely-generated modules

(All my rings are commutative with $1$.) Reworded a little, a question in a previous Commutative Algebra exam goes like this: Let $A$ denote a ring, $X$ denote an $A$-module $F$ ...
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44 views

Why is well order defined like that?

a well ordered set is a partially ordered set where every subset has a first element, but why not just say "every pair of elements in the set has a first element?" is there a more general reason to do ...
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111 views

Neighborhood chain in $\mathbb{R}^\mathbb{R}$

Edited after SamM's comment: Consider the topological space $\mathbb{R}^\mathbb{R}$, with the usual topology. Pick a point $x \in \mathbb{R}^\mathbb{R}$ and a neighborhood $V = V_0$ of $x$. I wish to ...
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Lattices and Boolean algebra

I have read in a text book that the set of natural numbers form a lattice under divisibility. How can it possibe, since there is no upper bound and therefore a Sup of the set?
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Hasse diagrams for sorts of size 3 and 4?

Does anyone know what does the following mean? I understand that a Hasse diagram represents a given partial order but I don't seem to get this example. Below is a Hasse diagrams for sorts of size 3 ...
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How can I prove a Well-found ordering in Set Theory

How can I prove the ordering function as following, Let $(x_1,y_1)\le(x_2,y_2) \overset{\text{def}}\Leftrightarrow x_1\le x_2 \land y_1\le y_2$ Prove that $(x_1,y_1)\le(x_2,y_2)$ over $\...
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Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is ...
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32 views

Why aren't there uncountably many disjoint open intervals of $\mathbb{R}$?

I know that this can't be true given that $\mathbb{R}$ is separable, but I'm having a hard time coming to grips with why this is, exactly. In particular, can't I just take some uncountable strictly ...
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Checking injectivity of a certain function from a union of a family indexed by $K$ to $K$

Let $A = \{ A_k | k \in K \}$ be a family of sets indexed by $K$. By Zermelo's theorem, $K$ can be well-ordered. Now, let $\leq$ be a well-order on $K$. Define $j: \bigcup\limits_{k \in K} A_k \to ...
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Hasse diagram of a total order not linear

A total order is also called a linear order. But the Hasse diagram of a total order does not need to be a simple single line. So the terms 'linear' and 'chain' are misleading. Is this correct?
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Does convergence of bounded, increasing sequences generalize from the set of real numbers to arbitrary partially ordered sets?

If $t$ is any real number, then I can find a strictly increasing sequence $(t_n)$ of real numbers converging to $t$. E.g. $t_n = t - \frac{1}{n}$ would do. Does this generalize to arbitrary partially ...
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83 views

Is [a, a[ empty?

Is the segment $[a, a[$ equivalent to the point $\{a\}$ or the empty set $\varnothing$? Can one or other be formally proved? I was wondering because in computer science it is the empty set, as the ...
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51 views

Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$?

Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$? (With lexicographical order) Thanks
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Example of non-way-below Auxiliary Relation in a Poset.

As the title have suggested, I am looking for an example of an auxiliary relation in a poset that is not the way-below relation. Any suggestion of books/paper to read is appreciated. Thanks. Edit: ...
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Order-preserving maps on a directed-complete poset

Definitons. A poset $P$ is directed if every finite subset has an upper bound in $P$. It is directed-complete if every directed subset has a least upper bound in $P$. For any poset $P$ let $[P\to P]$ ...
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A classification of the types of poset extensions that raise local-global principle is equivelent to the axiom of choice?

EDIT: Completely overhauled the question, remove category theory from it (stated in plain set theory): The original question Adding relations to a partial order Let $X$ be a set and let $P: Pow(X) \...
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Adding relations to a partial order

Let $(X,\leq)$ be a poset. For every $A\subseteq X$, $a \in A$ is called isolated iff there are only finite number of elements in $A$ that can be compared to it. A continuous (probably not the best ...
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25 views

Higher order partial order

I'm trying to find an effective and making sense way (for programming purposes, but also from math interest) to order partial orders, means, given a set and many partial orders (trees) over it. How ...
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37 views

Free cocompletion of thin categories

Let $C$ be a thin category. Is there a cocomplete thin category $C\to C'$ such that for every functor $C\to D$ into a cocomplete thin category there is a unique, up to isomorphism, cocontinous functor ...
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Partially ordering finite graphs

What are some interesting partial orders on the set of all finite graphs (identified up to isomorphism), apart from the usual (induced) subgraph relation and the (topological) minor relation, and why ...
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55 views

Flip/flop of finite joins and finite meets of lattices

Let $\mathfrak{A}$, $\mathfrak{B}$, $\mathfrak{C}$ be lattices (in fact in the example I have in mind, they are distributive and even co-Heyting lattices). Let maps $f:\mathfrak{A}\rightarrow\mathfrak{...
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Given a partial order $R$ on the set $A$, prove that there exists a total order $\leq$ on $A$ such that $R \subseteq {\le}$

Let $A$ be a finite set, and $\langle A,R\rangle $ be partially ordered. $\textbf{Prove}$ that there exists a total order $\leq$ over $A$ such that $R \subseteq {\le}$. $\textbf{My Attempt:}$ ...
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Filters and antichains in preorders

In a preorder $\mathbb{P}$ (reflexive and transitive), what means to say that a subset $C\subset\mathbb{P}$ is not contained in any filter of $\mathbb{P}$? For example, it is obvious that if $C$ ...