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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Proving that $\overline{([a,b],\le)}=\overline{([c,d],\le)}$, i.e., the order types of two closed intervals are the same

Prove that for all $a,b,c,d : a\le b : c\le d$ we have 1.$\overline{[a,b]}=\overline{[c,d]},$ 2.$\overline{(a,b)}=\overline{(c,d)}$ 3.$\overline{[a,b)}=\overline{[c,d)}$ ...
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Show that If $C$ is a chain in $X$ then $f(C)$ is also chain in $Y$.

Let X and Y are poset and $f:X\to Y$ is increasing function. If $C$ is a chain in $X$, show that $f(C)$ is also chain in $Y$. Since C is chain for every $x,y \in C: (x,y)\to \left(x\leq y\bigvee ...
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How to prove that $x\leq f(x)$ if f order isomorphic?

Let X is a well ordered set and $f:X\to f(X)=Y\subseteq X$ is order isomorphic. For any $x\in X$ prove that $$x\leq f(x)$$ since X is well ordered, {x,f(x)}$\subseteq$X and $x\leq f(x) $ or ...
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101 views

Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor Conjecture

Forcing notions are partial orders. In some sense each partial order is a "combination" of some well-orderings and each well-orderings is isomorphic to a unique ordinal number. Thus in some sense a ...
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61 views

Prove/disprove questions on isomorphism and embedding between order types

About the notations: Let $\lambda, q, z, \omega$ be the order types of the reals, rationals, integers and natural numbers respectively. The sign $=$ means there's isomorphism and $\le$ means ...
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67 views

Understanding Recurrence Relation

as i ask question and answered by some Clever people at this topic: Recurrence Relation Solving Problem i try to learn new thing with new question very similar to get familiar with recurrence ...
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68 views

Have you seen this property of tolerance relations before?

Let $A$ be a set equipped with a binary reflexive and symmetric relation $\uparrow$ (such relations are often called tolerances, see also "Are there real-life relations which are symmetric and ...
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36 views

Is the series $\frac{1}{(n+1)^p}-\frac{1}{(n-1)^p}$ where 0<p<1 convergent or divergent?

Sorry for my bad English. I really suspect it is convergent. But I can't prove it. Since ${x^p}$ is not derivable at x=0, I can't using taylor expansion to find the order of infinitesimal, thus ...
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Is there a mistake in this question: $\forall a\in A: |\{x\in A:x\le a \}|=|\{ y\in B :y\le a \}|$?

Two ordered sets $(A,\le_A), (B,\le_B)$ and there's an isomorphic function $f:A\to B$ Prove $\forall a\in A: |\{x\in A:x\le a \}|=|\{ y\in B :y\le a \}|$ I think there's a mistake in this ...
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Recurrence Relation Solving Problem

Can anyone help me in solving this complex recurrence in detail? $T(n)=n + \sum\limits_{k-1}^n [T(n-k)+T(k)] $ $T(1) = 1$. We want to calculate order of T. I'm confused by using recursion tree ...
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1answer
29 views

Total function vs Partial function

I'm reading a lecture note in which the following functions ($fact_i : \mathbb{Z}_\perp → \mathbb{Z}_\perp$) for $i \in \mathbb{N}$ are NOT considered total: \begin{equation} fact_0(x) = \perp ...
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28 views

Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single ...
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45 views

If a net is too big, convergence tells us very little?

I'm just learning about nets. It occurs to me that sometimes $x_{\alpha} \to x$ doesn't tell us much at all. Consider the directed set $J:= \{(U,x)\in \mathcal{P}(X) \times X: x \in U \}$, $(U,x) ...
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Is every continuous function measurable?

In non-Hausdorff topology it is standard to define the Borel algebra of a topological space $X$ as the $\sigma$-algebra generated by the open subsets and the compact saturated subsets. Recall that a ...
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41 views

Describing orders on sets $\{1,2,3,4\}$ and $\mathbb N$ with minimal/maximal/maximum/minimum elements

Describe all the partial orders on $\{1,2,3,4\}$ where the set of minimal elements are $\{2,4\}$ and the set of maximal elements is $\{1,3\}$ Describe all the partial orders on $\{1,2,3,4\}$ ...
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Has anyone used the isomorphism with $\Bbb{N}_{\gt 0}$ as a monomial ordering?

Let $R[x_1, x_2, \dots]$ be a ring of formal polynomials in a countably $\infty$ number of indeterminates $x_i$, over a commutative ring $R$. The commutative monoid $X = \{ x^e = x_1^{e_1} x_2^{e_2} ...
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31 views

What is the meaning of $<$ in a preorder?

Let $(P,\le)$ be a preorder, i.e. $P$ is a set and $\le$ is a relation on it that is reflexive and transitive. In this context for myself I can find two interpretations for the symbol $<$ 1) ...
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Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q ...
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Has an order with this property a special name?

If $a$ is an element in a preorder then you can eventually go 'a step back' (and repeat this) in the sense of finding an element with $b\leq a$ and not $a\leq b$. Is there a special name for ...
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Every admissible ordering on a countably infinite monoid is induced by an isomorphism onto $(\Bbb{N}, \cdot)$.

Let $fg$ mean the functional composition of functions $f, g$. Let $M$ be a countably infinite, commutative, multiplicative monoid and let $f : M \to (\Bbb{N}_{\gt 0}, \cdot)$ be an isomorphism. ...
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Continuity and Leximin

Consider a relation $\geq$ over the set of real-valued vectors. We say that $\geq$ is continuous if for any positive integer $n$, and any $\pi\in\mathbb Z^n,u\in\mathbb R ^n$ we have that the sets ...
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All tree orders are lattice orders?

Say that a set is tree ordered if the downset $\downarrow a =\{b:b\leq a\}$ is linearly ordered for each $a$. In a comment, Keinstein says that such sets are also semi-lattices, provided they are ...
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Addition on well ordered sets not-commutative by showing $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$

My goal is to show that addition on well ordered sets are non-commutative by showing that, $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$ Some definitions (let A and B be ...
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Number of join-irreducible elements of a lattice: is it monotonic?

Let $\mathcal L$ be a sub-lattice of $\mathcal P(X)$, where $X$ is a finite set. Denote by $\mathcal I(\mathcal L)$ the set of union-irriducible elements of $\mathcal L$ (i.e. $A\in \mathcal ...
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Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing a singleton set? [duplicate]

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
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54 views

What is $|Aut(D_n,|)|$?

Let $n=p_1^{\alpha_1}\dots p_k^{\alpha_k}$ with the $p_i$ distinct primes and $ \alpha_i\in \Bbb N$. Just to check if I'm correct, is it true that $k!$ is the number of order-isomorphisms of the form ...
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Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing singleton sets?

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
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Proving isomorphism in lattices

The question is as follows: Let f be a monomorphism from a lattice $L$ to a lattice $M$.Show that $L$ is isomorphic to a sublattice of $M$. My attempt: Since $f$ is a monomorphism from a lattice ...
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A problem concerning a partially ordered in $\omega$ and two chains.

Assume that $P=\left\langle\omega, \preceq\right\rangle$ is a partially ordered set such that for each $n \in \omega$ there are two chains $A_n$ an $B_n$ in $P$ such that $n \subset A_n \cup B_n$. ...
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extension of an increasing function over a lattice

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
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what kind of relationship is “is prefix of”?

Consider the "is prefix of" relationship on a set that corresponds to the words of some alphabet. E.g. "ab" is prefix of "abc". This relationship is: antisymmetric transitive reflexive ("ab" is a ...
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Finite set with $\sup\{S\} \notin S $

Just a quick question about sets. Can anyone here think of an example of a finite set $S$ where the $\sup\{S\} \notin S $ ? I found this on a past exam paper for college, and am unsure if there ...
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Hasse Diagrams - Relations

I have the following question: Draw the Hasse diagram for the following partially-ordered set: The relation $X$ is a subset of $Y$, on the set $\{ \{0\}, \{2\}, \{0,1\}, \{0,2\}, \{2,4\}, ...
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Existence of net of positive real numbers

Does for every totally ordered infinite set $T$ with no greatest element, there exist a net $(a_t )_{t\in T}$ of positive real numbers which is convergent to $0.$ ?
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(Almost) co-free objects?

another naming question from me, which comes up because I try to use category theory as a compass in developing some (new or not?) order-theoretic notions. According to my book (The Joy of Cats, ...
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Small posets with prescribed number of linear extensions

Given a natural number $n$, I want to construct a (finite) poset $P_n$ such that $P_n$ has exactly $n$ linear extensions. This can always be done, for instance taking $P_n$ to be a chain of length ...
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Introductions to posets on algerbaic structures (Everything I need to know about them)

I need a good and complete introduction to Tree-like orders and partial orders on algebraic structures with one operations. I accept basic texts too. I'm looking for free online texts mostly because ...
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total order on finite dimensional vector space over $\mathbb{R}$

We know that, If we start with a basis of a finite dimensional vector space $V$ over $\mathbb{R}$ by using lexicographic ordering with respect to that basis, we have a total ordering $\lt$ on $V$ ...
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Order of the subgroup <g>H of G

This is an old exam question I'm trying to solve: Having a group $G$ and $H$ a normal subgroup of $G$ with order $n$ and taking $g$ in $G$ to be such that $gH$ has order $m$ in $G/H$, I wish to prove ...
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Something interesting about partial orders to show to my students

I'm looking for some interesting examples or theorems about partial orders to show to my students, can be also some fundamental theorem such as an analogue of this theorem on equivalence relations. ...
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Exercise 1.17 from Bell & Slomson's Models and Ultraproducts

I'm attempting to prove the following theorem left as an exercise from Bell & Slomson's Models & Ultraproducts (1969). I'd like to know whether my attempted proof is correct, and if not, I'd ...
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$\mathbb{R}$ and $\mathbb{R}\setminus \{0\}$ are not isomorphic as linear orders.

Informal argument So, we know that $\mathbb{R}$ with its usual topology is connected, and of course $\mathbb{R}\setminus\{0\}$ is not. Any $``$supposed" isomorphism should grant us a homeomorphism ...
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The number of partial orders of finite set.

My question is about the number of partial orders of finite set. For example, if $S=\{a, b\}$, then there are 4 partial orders $\{ (a, a), (b, b) \}$, $\{ (a, a), (a, b), (b, b) \}$, $\{ (a, a), (b, ...
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Showing another result on distributive, complemented lattice

Currently reading Bell & Slomson's Models & Ultraproducts. Looking for clues, tips hints for showing the following: Let $L$ be a distributive, complemented lattice. Then for any $x$ $\in$ ...
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Given a collection of ordered sets, find minimal order-preserving superset

Say I have some collection of ordered sets $C = \{S_i\}$. Is there an efficient way to determine the minimal ordered set $S^*$ such that each of the original $S_i \subseteq S^*$, with order ...
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How many different proper subfields does $K$ have, where $K$ is a field of order$p^n$?

If $p$ is a prime, and $d$ is an integer greater than or equal to 1. Let $n= p^d$, then there exists $K$ being a field of order $p^n$. But how many different proper subfields does $K$ have? The ...
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Theorem on complemented lattices

I'm looking through the book Models and Ultraproducts by Bell & Slomson, and I'm checking that my proof is correct (exercise 1.13). I'm pretty sure it is, but I thought I'd post it anyway. It is ...
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Operation table of Hasse diagram

Consider the following Hasse diagram: My book gives the following join and meet operation tables for this diagram: $$\begin{array}{|c || c | c|} \hline Subset & x \wedge y & x \vee y \\ ...
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Do any familiar adjunctions arise from this construction?

I was pondering an answer that user Andreas Blass provided to an old question of mine and wondered if the following construction cropped up anywhere other than the example I will provide. For any set ...
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The length of a poset

Let $(X,\le)$ be a (nonempty) poset. The length of $(X,\le)$ is defined by $$\operatorname{len}(X)=\sup\{|C|-1\mid C\subseteq X \text{ is a chain} \}$$ Can $\sup$ be replaced by $\max$? (without ...