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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Order topology on a subset may be weaker (but never stronger) than the subspace topology

If $X$ is a linearly ordered set, the topology $\mathcal{T}$ generated by the sets $\{x:x<a\}$ and $\{x:x>a\}$ ($a \in X$) is called the order topology. Suppose $Y$ is a subset of $X$, show ...
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Characterisation of the order topology on a linearly ordered set

If $X$ is a linearly ordered set, the topology $\mathcal{T}$ generated by the sets $\{x:x<a\}$ and $\{x:x>a\}$ ($a \in X$) is called the order topology. If $a, b \in X$ and $a<b$, there ...
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Prove that $\mathscr F^+ = \mathscr J^+$ and $a \in \mathscr J^+$ iff there exists a filter

Let $\mathscr F$ a filter and let $\mathscr J = \mathscr F^c$ be an ideal dual to $\mathscr J$. Denote $\mathscr F^+ = \{a : (\forall x \in \mathscr F)[a \land x \neq 0]\}$ and $\mathscr J^+ = \{a:a ...
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A finite subset of an ordered set contains an $\inf$ and $\sup$ [duplicate]

Let S be an ordered set. Let A ⊂ S be a nonempty finite subset. Then A is bounded. Furthermore, inf A exists and is in A and sup A exists and is in A. Hint: Use induction. How do I use induction to ...
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Are the maps f(n)=3n and h(x) = the greatest natural number ≤ x residuated?

I have issues with proving the next things: Let $P = ⟨N,\le⟩$ the poset of the natural numbers with the standard order. Consider the map $f : N → N$ defined by $f(n) = 3n$. Is $f$ residuated? Let $P ...
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Exponentiation on order types

How is exponentiation defined on order types? We know that $2^\omega=\omega$. What is $2^{\omega^*}$? Is it $\omega^*$? $\eta$? $\lambda$? I'm guessing $\eta$, but I'm not sure. $\omega$ is the ...
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Are $((0,1]\cap\mathbb Q)\times\mathbb Q$ and $\mathbb Q \times([0,1)\cap\mathbb Q)$ order isomorphic?

The ordering here isn't specified but I assume it's lexicographic. I think the answer is yes. My reasoning is the following: Neither sets have a least or a greatest element because the second ...
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Universal set for linearly ordered sets of given cardinality

It is easy to see that every infinitely countable linearly ordered set embeds into Q (rational numbers) with its linear order.("Linear order"="total order". All embeddings are assumed to preserve ...
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Understanding proof of Hartogs’ Theorem on Set Theory

I'm trying to understand the proof of the Hartogs’ Theorem on page 100 of this book. My especific question is: If we have for each set $A$ $$\mathrm{WO}(A)=\{ (U,\leq_{U}) \, | \, U\subseteq A \, ...
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Proving lattice distributive property - discrete math [closed]

I'm studying discrete math. I'm stuck on this question. Thank you for solution. I don't have any idea to solve it. ...
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Condition to be a prewellordering.

I'm trying to do the problem 7.17 on th book Notes of Set Theory of Moschovakis. First I will define what is a prewellordering: A prewellordering on a set $A$ is any relation $(\lesssim) ⊆ A×A$ ...
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Example of a map between two lattices such that the map is an order embedding but not a lattice embedding

Is there an example of a map $h$ between two lattices such that $h: L_1 \rightarrow L_2$,that is order embedding from $L_1$ to $L_2$ but not an embedding when they are taken as algebras? I would ...
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Question about maps in partially ordered sets

I am struggling with a proof related to the mapping of posets. Let $P_1$and $P_2$ are posets, and $f$ an order preserving map from $P_1$ to $P_2$. $$g(Z):= f^{-1}[Z]$$ $g$ goes from the set of all ...
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Prove that every totally ordered set has a well-ordered cofinal subset

I thought this was easy until I realized that for a set to be well-ordered, EACH of it's non-empty subsets must have a least element. Let $A$ be a non-empty totally ordered set. Let $x_0$ be some ...
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What is the relationship between the cofinality and well orders?

Suppose $A$ is an infinite linear order with cardinality $\kappa$, and take the family $\langle A_{\alpha}: \alpha < \kappa \rangle$, define like this: $$A=\bigcup_{\alpha < \kappa} ...
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Partial order of a finite set is an intersection of a finite number of linear orders of this set

I am trying to prove that every partial order of a finite set is an intersection of a finite number of linear orders of this set. Can this be proved using these observations: a partial order has a ...
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Equivalent forms of axiom of choice

I'm currently reading Dugundji's Topology book. Exercise 2 of section 2 in page 58 gives 4 equivalents of Zorn's lemma. I'm only interested in a) and c) (I won't copy the whole thing cause it's too ...
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Determine which of these sets is a lattice

I'm currently reading Dugundji's Topology's book. Exercise 4 in chapter 2 says (omitting a part that is irrelevant to the question) A partially order set is a lattice if each pair of its elements ...
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Are $[0,1]\times[0,1]$ and $[0,1]$ order isomorphic?

The ordering on the first set is reverse lexicographic and the order of the second set is the usual ordering of reals. My intuition tells me that the answer is no. The only isomorphism between those ...
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Total order function property

This may have been asked before but it's difficult to search for. Suppose that $|A|=n$ and that $R$ is a total order on $A$ (not a strict order). Define $g:A \to I_n$ by $$ g(a) = |\{x \in A ~|~ ...
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Least finite linear orders with same theory in monadic second order logic.

Today I want to ask a relaxed version of my last question. So if somebody finds a solution to that question he will immediately get a solution for this question here. Question. Let $m<\omega$ ...
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When do finite linear orders have the same theory in MSO?

Let $\mathfrak A$ and $\mathfrak B$ be finite linears orders and $m<\omega$. Then we have $$\mathfrak A \equiv^m_{FO}\mathfrak B \quad\text{iff}\quad |A|=|B| \,\, \text{or}\,\, |A|,|B|\ge 2^m-1.$$ ...
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Counterexample to the Hausdorff Maximal Principle

The Hausdorff Maximal Principle states: Every partially ordered set $\left(X,\leqslant\right)$ has a linearly ordered subset $\left(E,\leqslant\right)$ such that no subset of $X$ that properly ...
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Mathematical structures with a canonical partial order

I like order theory very much, and am always excited to discover mathematical structures that can be intrinsically endowed with a partial order. "Easy" examples are $T_0$ topological spaces (with $x ...
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Is it an axiom that the inequalities are translation-invariant or can we prove it?

I was thinking about the inequalities on the set of real numbers. To me and everyone else, it's been taught that an inequality is translation-invariant, i.e.: $x < y \implies x + c < y + c ...
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Order preserving injection from first uncountable ordinal to the real numbers

Can there be an ordering preserving (1-1) map $f \colon \langle\aleph_1,\in\rangle \rightarrow \langle\mathbb R, \lt\rangle$? I remember that if $f$ is order preserving then consider the following ...
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Largest, smallest, LUB and GLB of the relation $\leq$on $ \left\{\frac{2n + 1}{n + 1} \mid n \in \mathbb{N}\right\}$

I have to fin the largest, smallest, lower/least upper bound (LUB) and greatest lower bound (GLB) element of the relation $\leq$ on the following set: $$S = \left\{\frac{2n + 1}{n + 1} \mid n \in ...
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Number of upward closed subsets

I am looking for an algorithm of some sorts that can produce the total number of upward closed subsets in a partial ordered set. Let $\langle P,\leq\rangle$ be a (finite) partial ordered set, then an ...
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How to sort 4 (or more) values with non-piecewise functions?

This question was inspired by this: Finding a non-piece wise function that gives us the $i$'th largest number. My question is how to do this for four or more values. In other words, given 4 ...
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Order type of real analytic monotonic functions ordered by eventual domination

Let $\mathcal F$ be the set of all functions $\mathbb R^+\to\mathbb R$ that are: real analytic on $\mathbb R^+$, monotonic on $\mathbb R^+$, and having derivatives of any order that are also ...
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If $R$ is a strict partial order prove that is asymmetric

Suppose $R$ is a relation on a set $A$, and $R$ is asymmetric if: $\forall x \in A$ $\forall y \in A$ $((x, y) \in R \rightarrow (y, x)\not \in R)$ The first point of the exercise was to demonstrate ...
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Similarities when adding a new point to a poset.

I'm trying to solve the problem 7.3 of the book Notes on Set Theory written by Moschovakis. Basically, I have to prove that for every poset $P$ we have $\mathrm{Succ}(P)=_o P+_o [0,1)$, were ...
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Why $B=\{\frac{1}{n} \mid n \in \mathbb{Z^{+}}\}$ does not have a smallest element?

I am studying from some slides where there's written that $B=\{\frac{1}{n} \mid n \in \mathbb{Z^{+}}\} = \{1,1/2,1/3,1/4, \mid\}$ does not have a smallest element but it does have a g.l.b (infimum) ...
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Minimals in partially ordered sets

In a subset $Y$ of a partially ordered set $(X, \leq)$ a minimal element is an element $a \in Y$ such that $(\forall y \in Y)\ y \leq a \Leftrightarrow y = a$ However in this diagram ...
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A question about $\aleph_1$-dense sets and the basis problem for uncountable linear orderings

I have a question which I have been unable to find a reference for and which I explain as follows: Recall that a set of reals $X$ is $\kappa$-dense if between any two real numbers there are exactly ...
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Lattice of interval

I would like to create a lattice of intervals of integers, but I don't know how to 'draw'(Hasse diagram) it. An interval looks like: [1,4] or ]-Inf,3] for example. I'm having difficulties deciding ...
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partial order, sets

Please help to solve the following: The set is partially ordered with respect to the “less than or equal to” relation, ≤, for real numbers. In following case, determine whether the set has a ...
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Hasse diagram for the divisibility order on $\{2,3,4,6,8,9,12,18\}$

Please help with the diagram, as I am not sure if I draw the right one and I have got two different results. Please see screen shot attachment of the drawing. Consider the “divides” relation on each ...
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Are there general conditions under which minimal generating sets can be expected to exist?

There exist algebraic structures $X$ with no minimal generating set. For example, $\mathbb{Q},$ viewed as an Abelian group. There also exist algebraic structures whose every generating set includes ...
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Subsets of the rationals order isomorphic to transfinite ordinals

Let $\langle\mathbb Q,\lt\rangle$ be the set of rational numbers with the usual $\lt$ ordering. Find subsets $A$ and $B$ of $\mathbb Q$ which, with the usual ordering, are order isomorphic to the ...
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For which countable successor ordinals $\alpha$ is the reverse order isomorphic to the ideals of a PID ordered by inclusion?

Let $\alpha$ be a countable successor ordinal and $\alpha^{\mathrm{op}}$ the reverse order. For which $\alpha$ is there a commutative principal ideal ring $R$ such that the ideals of $R$ form a chain ...
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Does Green's $\mathcal{J}$-relation define a total order on the equivalence classes?

In a semigroup $S$ we define $a\le_\mathcal{J} b$ iff $a=xby$ for some $x,y\in S^1$. Defining $a\equiv_\mathcal{J} b$ by $a\le_\mathcal{J} b$ and $b\le_\mathcal{J} a$ gives a partial order on ...
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Basic question on set ordering

I have some difficulties with a question from Jech's book (Introduction to Set Theory). Any help would be appreciated. Give examples of a finite ordered set $(A, \leq)$ and a subset $B$ of $A$ so ...
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Is there a universal mapping property satisfied by the disjoint union of totally ordered sets?

It is proved in another post that the product and coproduct do not exist in the category of totally ordered sets (except in some trivial cases). (In this post I will only consider the category TOrd, ...
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Scott continuity on powerset

I am looking for the name of the class of functions $f:\mathcal P(A)→\mathcal P(A)$ that are monotone and that are characterised by their image on finite subsets, i.e. the functions $f$ satisfying the ...
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Origin of min/max notation

Here I am referring to the notation $x \wedge y = \min \{ x,y \}$ and $x \vee y = \max \{ x,y \}$. These seem to reference the corresponding usages in logic, where $\wedge$ means "and" and $\vee$ ...
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Statements comparable with Axiom of Choice in ZF

Let $AC$ denote any fixed statement of the Axiom of Choice in $ZF$. Consider the set of statements $\phi$ in the language of $ZF$ such that either $ZF+\phi$ proves $AC$ or $ZF+AC$ proves $\phi$. The ...
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Name of an axiom

What's the name of the following axiom: Let $A$ and $B$ be sets of real numbers, and $(\forall a\in A\text{ and }\forall b\in B)a \leq b$. Then $\exists c\in\mathbb R$ such that $(\forall a\in A\text{ ...
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Definition of ordered ring/field

One way to define an ordered ring is as a ring with a total order $\leq$, satisfying $x\leq y\implies x+z\leq y+z$; $0\leq x$ and $0\leq y\implies 0\leq xy$. Does it hurt to replace 1. with the ...
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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 ...