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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Can every poset be toplogically sorted?

A partial ordering $\preceq'$ on $S$ is said to be compatible with another partial ordering $\preceq$ if for all $a,b,\in S$, $$ a \preceq b \Rightarrow a \preceq' b$$ Given a poset $(S, \preceq)$, ...
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Any efficient approach to find the minimal vectors?

$\bullet$ A vector $X=(x_1,\cdots, x_m)$ is less then vector $Y=(y_1,\cdots,y_m)$ when $x_i\leq y_i$, for each $i=1,\cdots, m$, and for at least one $j$, we have $x_j<y_j$. $\bullet$ A vector $X\...
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Classifying singularities and determing orders of complex functions

Here are a few functions for reference purposes: $f(z) = \frac{sin(2z)}{z^3}$, $ \space g(z) = \frac{sin(z)}{tan(z)}$, $ \space h(z) = z^2 sin(\frac{1}{z})$ Suppose I was calculating the ...
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Characterization of noncommutative $L^2$-spaces as ordered vector spaces

If $M$ is a von Neumann algebra and $\tau\colon M_+\to[0,\infty]$ is a normal, semi-finite, faithful trace, the associated GNS Hilbert space is the completion of $\{x\in M\mid \tau(x^\ast x)<\infty\...
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Is BPIT equivalent to some ordering principle?

Working in $\mathsf{ZF}$, is $\mathsf{BPIT}$ (Boolean Prime Ideal Theorem) equivalent to some statement of the form "every set can be ***ly ordered"? I know that $\mathsf{BPIT}$ implies that every set ...
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“Contiguity” classes of order morphisms

I was having problems with this exercise that seems really hard. Let $X$ and $Y$ be both FINITE posets and $f$, $g:X \rightarrow Y$ order morphisms such that $f \leq g$. Then there exists $f_0, f_1, .....
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Finding distance from valid number

I have a game related problem that is pretty complex. Here is the simplified version of the problem. I have a list of "good" numbers. ...
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If $P$ has all binary joins, and all chain-shaped joins, is $P$ necessarily a complete lattice?

Question. Let $P$ denote a poset with all binary joins, and all chain-shaped joins (of arbitrary cardinality). Is $P$ necessarily a complete lattice? Motivation. Let $f$ and $g$ denote closure ...
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Every $\alpha < \kappa^+$ can be embedded in any interval of $\kappa^{<\omega}$.

Let $\kappa$ be a cardinal. I want to show that every ordinal $\alpha < \kappa^+$ can be embedded (as an order) in any interval of $\kappa^{<\omega}$, ordered lexicographically. This is exactly ...
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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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Can an ordinal be limit of a smaller set of ordinals?

The Wikpedia article Regular cardinal contains the following weird sentence: An infinite ordinal $\alpha$ is regular if and only if it is a limit ordinal which is not the limit of a set of smaller ...
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No uncountable subset can be well-ordered $\iff$ Hartogs ordinal is $\omega_1$

I'm looking at an exercise which starts as follows: Suppose that no uncountable subset of $\mathcal P\omega$ can be well-ordered (equivalently, the Hartogs ordinal $\gamma(\mathcal P\omega)$ is $\...
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Product of directed partial orders

Is a product poset (with componentwise order) of nonempty posets a dcpo if and only if each multiplier is a dcpo? (for both binary and arbitrary products)
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Question about a linear order

Let $L$ be the set of countable limit ordinals. For each $\alpha \in L$, let $\langle \alpha_n : n < \omega \rangle$ be a strictly increasing cofinal sequence in $\alpha$. Define a linear order on $...
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Is there any well-ordered uncountable set of real numbers under the original ordering?

I know that the usual ordering of $\mathbb R$ is not a well-ordering but is there an uncountable $S\subset \mathbb R$ such that S is well-ordered by $<_\mathbb R$? Intuitively I'd say there is no ...
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Get well-order from total order by choosing finite subsets

Exercise 7.9 in this book: Given that any set has a multiple-choice function, ie a function picking out a nonempty finite subset of each nonempty subset, show that any totally orderable set can be ...
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(Partial/total/well-) order vs ordering

Order or ordering – what is the difference? Is either correct? Is one British and one American English? Not exactly a maths question but probably still the best place to ask.
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How do I prove that every chain has an upper bound?

Let $A$ be a non-empty set. Let $X$ be the collection of bijections $f:U→V$ where $U,V$ are disjoint subsets of $A$. Define the relation $≥$ as follows: $$(f:U→V) ≥ (f′:U′→V′) \text{ iff } U′⊆ U \...
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Is directed set countable, if for each element there are only finitely many smaller ones?

A directed set is a pair $(A,\leq)$ where $\leq$ is a reflexive, transitive relation such that for any $x,y\in A$ we have some $z$ such that $x,y\leq z$. (This comes up when dealing with categorical ...
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$ord(f(a)) \leqslant ord(a) $ for every a in G

Let $f: G \to H$ be a group homomorphism. I want to show that: (i) $ord(f(a)) \leqslant ord(a) $ for every a in G (ii) If $ord(a)$ is finite, then $ord(f(a))$ divides $ord(a)$ (ii) If $f$ is an ...
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Is the equivalence of the ascending chain condition and the maximum condition equivalent to the axiom of dependent choice?

Assuming the axiom of dependent choice, for a partially ordered set $(X,\le)$, the following statements are equivalent: $X$ fulfils the ascending chain condition, i. e. every chain $x_1\le x_2\le\...
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Hausdorff Maximal Principle

"Hausdorff's Maximal Principle" says that any partial order P has a maximal chain (chain = linear suborder). It is equivalent to the axiom of choice. If we restrict Hausdorff's Maximal Principle to ...
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Prove that $R$ is a total order

Prove that $R$ is a total order $$R=\{(x,y)\in \mathbb{R}\times \mathbb{R}~|~ x\geq y\}$$ I just need to figure out the final portion for total order. I already have it at partial order. Thanks for ...
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Name for a “layered” type of partial order?

I have a partial order $\prec$ over a (finite) set $S$ satisfying the following property: There exists a function $f:S\rightarrow \mathbb N$ such that $x\prec y \Leftrightarrow 0<f(x)< f(y)$....
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Lexicographic ordering on ${\cal P}(\kappa)$

How is the "lexicographic ordering" (which is supposed to be a total ordering) on ${\cal P}(\kappa)$ defined, where $\kappa$ is any cardinal? (I apologize if this question is a bit fuzzy.)
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Let a and b be commuting elements in a group G such as $a^m=b^n=e$. Prove that $(ab)^{mn}=e$

The problem I am working on says, "Let a and b be commuting elements in a group $G$ such that $a^m=b^n=e$. Prove that $(ab)^{mn}=e$" So what I know from this information is that ab=ba, a is an ...
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Binary relation defined by order isomorphism?

Let's say I have $(E, \leq_{E})$ and $(F, R)$ where $\leq_{E}$ is an order on $E$, and $R$ is some binary relation defined over $F$. Now if I have a bijection $f : E \to F$ such that, $$\forall x, y ...
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Ordered ring and mutiplicative ordinal

If $(E,<)$ is a linear order, let $s(E)$ denote the supremum of the set of ordinals which (order-)embed in $(E,<)$. $s(E)$ is also the set of ordinals which embed in $E$ with a non cofinal range....
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Infinite Trees, Infinite Branches, And Choice

https://en.wikipedia.org/wiki/Tree_(set_theory) Defines a tree T order-theoretically as a poset that has a minimum element R, and for each t in T, the set of lower bounds of T is well-ordered. This ...
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Are there results for relations between upward and downward closed partitions of some powerset?

I stumbled upon this, given some set $X$ and its powerset $\mathcal{P}(X)$ and some incomparable set $\mathbb{S}\subseteq\mathcal{P}(X)$, i.e. for any $S,S'\in\mathbb{S}$ we have $S\setminus S'\neq\...
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Definability in $\Bbb N$ + $\Bbb Z$

Which elements are definable in $\Bbb N$ + $\Bbb Z$? Where an element, a, is definable if there exists a formula such that $\forall x(\phi(x) \rightarrow x = a) $. I have that all elements of $\Bbb ...
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How to arrange the following sets?

Given the set $\mathcal{P}(\mathbb{N})$ for the following order: For $A,B \in \mathcal{P}(\mathbb{N}) $ applies $A \leq_{set} B \Longleftrightarrow_{def} A = B$ or$ ~$ min$(A\triangle B)\in B$ We ...
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A non-separable, connected ordered topological space with no interval order isomorphic to the reals

Is there a non-separable, connected ordered topological space such that for each point $x$ in that space, there is no interval containing $x$ that is order isomorphic to real line?
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If there are order preserving injections between two countable sets, will there be an order preserving bijection?

During my Topology class the following question was brought up: Question: Suppose that $A$ and $B$ are ordered countable sets. If there are two order preserving injections $f:A \to B$ and $g:B \to A$,...
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If there is a simple group of order less than 36, then it must have prime order

I want to show that if there is a simple group of order less than 36, then it must have prime order. Is there a quick way to show this or do I have to go through each order 1 though 36 showing that ...
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Separative forcings add new sets

Let $\mathbb{M} = (M, \in)$ be a class which models ZFC. A forcing $\mathcal{P} = (P, \leq)$ on $M$ (where $P \in M$ and $\mathbb{M} \vDash P \in M$) is separative if for every $r \in P$ there are ...
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A map between two totally ordered sets is a homeomorphism if and only if the map is oreder preserving

I'm a little stuck with this problem and I'd appriciate any advice or hint you could give me: Let $(X,\leq)$, $(Y,\preccurlyeq)$ totally oredered sets and $\psi: X \rightarrow Y$ a function. Then: $\...
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intersection on Poset Topology.

Consider $(X,\prec)$ partial ordered set and consider the collection of set $\{U_L(x) \}_{x \in X}$ where, $U_L(x) = \{y\in X: y \prec x \}$ . It is easy to prove this is a base for some topology over ...
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Mobius functions for acyclic categories, question about formula in “Combinatorial algebraic topology” by Kozlov,

Let $C$ be an acyclic category with a terminal object $t$, in "Combinatorial algebraic topology" by Kozlov he defines Mobius functions for acyclic categories, he first starts by defining a function $\...
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Minimum number of chains in a partially ordered set

Imagine we have a partially ordered set like $(X,R)$. So, a set like $X$ is partially ordered by a relation like $R$. We define $\delta(X):=$minimum number of chains in $X$ which covers poset My ...
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Are there any infinite (virtually) polycyclic groups with lattice orders that are not linear orders?

I am interested in noetherian group algebras, so I am learning about polycyclic groups. Specifically, I want to generalize some ideas that work well with $k[\mathbb{Z}^n]$ utilizing the lattice ...
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Can someone explain the phrase “order-preserving permutation of a totally ordered set”?

If a set is totally ordered, then any nontrivial permutation would destroy that order, so I'm clearly missing something. I have seen this exact phrase in many papers and books, mostly those dealing ...
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Does every nonempty well-ordered set have a least element?

I was making my way through The Art of Computer Programming, when some words got me puzzled. Here is how the book describes the principle of well-ordering: Let "$\prec$" be a relation on set $S$, ...
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Strictly increasing function from $\alpha$ to $\aleph_0$

Given that $\alpha < \aleph_1$, does there exists a function $f: \alpha \rightarrow \aleph_0$ that is a strictly increasing function? This seems like it would be true since $|\alpha| = \aleph_0$, ...
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Antitone Galois connections, composition

Is composition of two antitone Galois connections defined? What are all "possible" ways to define composition of two antitone Galois connections?
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Order of Galois connections between two boolean lattices

Is the poset of Galois connections between two boolean lattices itself a boolean lattice? If not, does it hold for: complete boolean lattices? atomic boolean lattices? atomistic boolean lattices?
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Galois connections between boolean lattices - an alternative representation

Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) boolean lattices (with lattice operations denoted $\sqcup$ and $\sqcap$, bottom element $\bot$ and top element $\top$). I call a boolean funcoid a ...
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Two alleged counterexamples (about boolean algebras)

Trying to solve this question, I propose two possible counter-examples. Please help me to understand whether these cases are really counter-examples. Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) ...
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Linearly extend an induced subposet

Suppose $P$ is a finite poset with partial order $\le_P$ and $Q$ an induced subposet, its partial order being ${\le_Q} = {\le_P}\cap Q^2$. Suppose we linearly extend $\le_Q$ to a linear order $\...
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Boolean lattices vs boolean rings

Which kinds of theorems about boolean algebras are easier to prove with boolean rings (than with actual boolean lattices)? Give me at least one example, as an answer.