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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Q: Trichotomy of order of the real numbers

I am currently reading Terence Tao's "Analysis I" and while progressing through the book, the reader is repeatedly asked to prove trichotomy properties of order for the natural numbers $\mathbb{N}$, ...
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1answer
49 views

Is every totally ordered finite dimensional vector space a lexicographic order for some basis?

Let's say we have a finite-dimensional vector space $V$ over a totally ordered field $\mathbb{K}$. Is every choice of totally ordered vector space structure (i.e compatible with the addition and ...
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1answer
21 views

Compact elements in a lattice satisfying ACC

I am trying to prove that in a complete lattice $L$ satisfying ACC (ascending chain condition), all elements are compact. To do so, I start taking $x\in L$ and $S\subseteq L$ such that $x\leq \...
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If a Galois connection does exists, how is it called?

Let $\phi$ be a function from a poset $B$ to a poset $A$. $f \mapsto \min \{ g\in B \mid \phi(g) \geq f \}$ is called the lower adjoint of $\phi$ and $\phi$ is called an upper adjoint. These two ...
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24 views

Plane partitions of a poset with one specified value

Given a poset $P$ and an element $x \in P$. How many plane partitions of height $m$ (order preserving maps from $f:P \to [1,m]$), exist when $f(x)=j, 1 \leq j \leq m$? I'm interested in this as a way ...
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21 views

Difference between maximum and maximal element of a set [on hold]

Does a set can have two or more maximum elements If so please give me explain and give example
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18 views

Trimming down a relation to a partial order

Is there any general approach for trimming a reflexive relation to a partial order on a finite set? In particular, given a relation R defined on a finite set, which is reflexive xRx, and for all x $\...
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2answers
31 views

Well ordering of the subsets of a given set

For a given set, does there always exists a well-ordering of the set of all its subsets which is stronger than the usual ordering (that is set-theoretic inclusion) of the sets of the subsets of the ...
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16 views

Intersection of converse well-founded quasiorders

Let $\mathcal{O}$ be a family of reflexive transitive relations on a set $A$, such that, for each $\preceq\in\mathcal{O}$ and each non-empty subset $B$ of $A$, the set $ \{ b\in B \mid \forall b'\in B ...
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18 views

Name for a directed set that is also a partial order but not necessarily a semilattice?

Is there a term for a structure $(S,\preceq,+)$ where $+$ is commutative, idempotent, and monotone with respect to $\preceq$? That is, for all elements $s$ and $t$, $s+t=t+s$, $s+s=s$, and $s\...
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42 views

Partial order on the set of ordinal functions

Let $\kappa$ be a regular ordinal. Say that for $f,g: \kappa \rightarrow \kappa$, $f \leq g$ iff $f(\alpha) \leq g(\alpha)$ for sufficiently large $\alpha$. Now define $(X_{\kappa},\preceq)$ as the ...
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1answer
15 views

Is reverse lexicographic order the same as graded reverse lexicographic order?

I want to make sure whether the two monomial orderings are actually the same thing. I am confused because the Cox book on Ideals, Varieties and Algorithms mentions only the graded reverse ...
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66 views

Categorical Interpretation of Strongest/Weakest Topology

One way to define the product topology is as the weakest topology for which all projection maps are continuous. Strongest/weakest topologies satisfying a given property are ubiquitous in topology and ...
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1answer
53 views

Uncountable family of maximal antichains whose union is countable

Given a ccc preorder $\mathbb{P}$ and a family $\mathcal{A}=\{A_\alpha:\alpha<\omega_1\}$ of pairwise distinct maximal antichains of $\mathbb{P}$, is it possible that $|\bigcup\mathcal{A}|\leq \...
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1answer
17 views

How to read partial ordering in a set?

Let $X$ be a partially ordered set with partial order $\preceq$. Then how can we read $x\preceq y$. Is it $x$ less than or equal l to $y$.?
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1answer
11 views

What is an example of an upward directed set that is not a join semilattice?

I understand that join semilattices are upward directed sets, but why not the converse? A simple counterexample would be very helpful.
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2answers
33 views

Difference between Ordering and Order?

I am confused by the two terms order and ordering. I am learning on Ideals, Varieties and Algorithms by Cox et all. The context is monomial orderings and Gröbner basis on polynomial rings. How are ...
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23 views

To prove: a product of profinite groups is again profinite.

I got stuck on this syllabus about abstract algebra: Let $J$ be a set and $\{ \pi_J \ : \ j \in J\}$ be a collection of profinite groups. Show that $\prod_{j \in J} \pi_j$ is again profinite. ...
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42 views

Maximal chains in posets under homomorphisms

Suppose that $P$ and $Q$ are two posets and $f:P\to Q$ is a homomorphism (a.k.a., $f(x)\le f(y)$ whenever $x\le y$). Given a chain $C\subset P$, the image $f(C)$ automatically is a chain as well. ...
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1answer
39 views

Proof of Ultrafilter lemma with two propositions and Zorn lemma

I would like to prove the following: Let $X$ be any set, then every filter $\mathcal{F}$ on $X$ is contained in an ultrafilter $F$ Using two propositions and Zorn Lemma. I am required to come ...
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1answer
52 views

is the product of totally ordered sets is total order [closed]

I was working in products of structures and I am trying to find a counterexample to the following: "the product of totally ordered sets is a totally ordered set." Unfortunately I could not find one. ...
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2answers
61 views

What are example of minimal but not least element? [duplicate]

I know that every least element are minimal. But, What are example of minimal but not least element?
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25 views

How do you prove that the greatest element of a partially ordered set (A,<=) is maximal? [duplicate]

1) How do you prove that the greatest element of a partially ordered set (A,<=) is maximal? 2) What will be some example that the maximal is not necessarily a greatest element?
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21 views

For a graded poset, why do we only consider the characteristic polynomial defined in terms of $\mu$?

When we have a graded poset $P$ with $0$ and $1$ we can define the characteristic polynomial $f_P(t)$ of $P$: $$f_P(t)=\sum_{x\in P}\mu(0,x)t^{r(1)-r(x)}$$ However, given a poset, have two functions, ...
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1answer
51 views

Any algorithm in literature to solve my problem? Should call it “Segment ordering”?

A slot with length $L$. I have $n$ segments, they have a positive integer length $x_1, x_2, \cdots, x_n$, respectively, and $\sum\limits_{i=1}^n x_i = L$. My goal is to fill the slot with these ...
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1answer
17 views

Sandwiched sequence converge to same limit in $\omega_1$

I am stuck on a question that might need a trick to crack, any help is appreciated Problem statement Let $(a_n), (b_n)$ be sequences on $\omega_1$ as a topological space, such that $a_n \leq ...
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2answers
58 views

Is it possible to have a linear order that is not “on a line”?

I am looking at some problems on linear order. It seems in all the problems, I am dealing with things that are 1D Whether it is $\mathbb{R}$ itself, or $\left\{\dfrac{1}{n}|n \in \mathbb{Z}_+\right\}\...
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1answer
48 views

Is it true that no linear order can have CCC?

I am reading about countable chain condition on Wikipedia https://en.wikipedia.org/wiki/Countable_chain_condition I want to quickly verify that if $(X, \leq)$ is a linear order, then it cannot ...
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1answer
57 views

Incompressible countable Total-ordering implies well-ordering i

A totally-ordered set (S,<) is incompressible if $(S,<) \cong (T,<)$ and $S \supseteq T$ implies $S = T$. Is it true that if $S$ is incompressible countable and totally-ordered then $S$ is ...
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67 views

Is there a concept of |x|<0? [closed]

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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175 views

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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35 views

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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1answer
41 views

Are $(\mathbb R\times \mathbb Q, \le_h)$ and $(\mathbb Q\times \mathbb R, \le_h)$ isomorphic? [closed]

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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92 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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3answers
51 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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3answers
90 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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0answers
17 views

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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47 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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58 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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3answers
25 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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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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0answers
10 views

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
58 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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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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1answer
40 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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1answer
15 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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2answers
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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1answer
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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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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 ...