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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Prove there exists a strictly increasing function from the natural numbers to a partially strictly ordered set

Let $P$ be a non-empty partially strictly ordered set and assume no element of $P$ is maximal (i.e. for every $x \in P$ there exists $y\in P$ with $x < y$). Show there exists a function ...
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Partial order relations and posets

Suppose you are given to prove this question: Let $P=\{2,3,4,\ldots,10\}$. Define $\le$ by $a\le b$ if and only if $a\mid b$, i.e, $a$ divides $b$. Prove that it is a partial order on $P$. I think ...
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$p$-adics with the least upper-bound property

It is well-known (I believe?) that the $p$-adics do not admit an ordering in the 'usual sense', the "usual sense" being a total order that is compatible with the field operations. I do not want to ...
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A chain with more than two elements

A chain with two elements $0$ & $1$ is complemented as complement of $0$ is $1$ and that of $1$ is $0$.How to show that every chain with more than two elements is not complemented?
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Order types of subsets of the reals

Is there a subset of $\mathbb{R}$ that has the order type $\omega + (\omega^*+\omega)\cdot\eta + \omega^*$ (i.e., an $\omega$-sequence, followed by a bunch of copies of $\mathbb{Z}$ ordered like ...
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Help Showing a Relation is/isn't a Partial Order

Define the relation $\le$, as $(a,b)\le(c,d)$ if and only if $a+b\le c+d$ and $a\le c$. Is this a partial order? I know it's definitely not if we remove the $a\le c$ (because then it's not ...
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Partial Ordering and Hasse Diagram.

Draw the Hasse diagram for the partial ordering “x is a factor of y” on the following sets: S = {2, 3, 5, 7, 21, 42, 105, 210} I don't know how to find the partial ordering of this set. I know that ...
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Restriction of a monotone self-map (“endomorphism of $\mathbf{Pos}$”) to a wide subposet

Here's a poset, call it $P.$ $\hspace{1cm}$ Define a function $S : P \rightarrow P$ as follows. $$S(0) = 1, \qquad S(1) = 2, \qquad S(2) = 2$$ Clearly, $S$ is monotone. Now consider the following ...
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How to prove this theorem? (Logical symbols help)

This is a theorem from a book. I'm having a hard time on proving it. Suppose A is a set,$\mathcal{F}\subseteq \mathscr{P}(A)$, and $\mathcal{F} \neq \emptyset$. Then the least upper bound of ...
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How many ways to order vectors in $\{0,1\}^n$?

How many different rankings can be produced for the vectors in $\{0,1\}^n$ that also respect the usual $\geqq$ ordering of vectors (defined below)? I want to produce a complete ordering where, for ...
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does a lattice have a minimal item

A lattice (L, ≤) is a partially ordered set, where each two elements of the set have a least upper bound and a greatest lower bound. Let's consider the situation where L is finite. I think that this ...
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Can I embed $\mathbb R^{\mathbb N}$ with a partial order into $^\ast\mathbb{R}$ with the linear order?

Define a relation $\prec$ on $\mathbb R^{\mathbb N}$ as, For all $f, g \in \mathbb R^{\mathbb N} $, $f \prec g$, if for all $n \in \mathbb N$, $f(n) \leq g(n)$, and there exists a $m \in \mathbb ...
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Prove that the greatest lower bound of $F$ (in the subset partial order) is $\cap F$.

This is one of the question I'm working on: Suppose $A$ is a set, $F \subseteq \mathbb{P(A)}$, and $F \neq \emptyset$. Then prove that the greatest lower bound of $F$ (in the subset partial ...
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About characterization of atomistic posets

Fix some poset. Let $\mathscr{A}$ be the map from elements of our poset into the set of atoms under this element. Is it true, that injectivity of $\mathscr{A}$ implies that our poset is atomistic? ...
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Prove if $x$ is greatest lower bound of $U$ then $x$ is the least upper bound of $B$

This is one of the problem I have been solving from Velleman's How to prove book; Suppose $R$ is a partial order on $A$ and $B \subseteq A$. Let $U$ be the ...
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Order-preserving map of regressive functions on $\omega_1$

I posted the following question a year ago on MO. It did receive some attention, but the answer there remains incomplete (as discussed in some detail in its comments). Meanwhile I got the impression ...
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ordering two rational functions

I have read that $$\frac{x^2 +3}{2x+1}$$ is less than $$\frac{2x-1}{2x+1}$$ in an ordered field,in $\mathbb{Q}((x))$, but how is that result computed? How do we compare two rational fractions like ...
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Infinite palindromes in number of nonisomorphic posets is independent of $\mathsf{ZF}$

The following is an "exercise" in P. Stanley's book "Enumerative Combinatorics": Let $f(n)$ be the number of nonisomorphic $n$-element posets (...) let $\mathcal{P}$ denote the statement that ...
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Birkhoff's definition of semilattice?

could anyone provide me with the original definition of semilattice by Garrett Birkhoff in his book on lattices? If you could also provide, page number and edition, it would be great (as well as some ...
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Concept of Boundedness

I noticed there are two notions of boundedness, one in the context of order theory and other in the context of metric spaces. In a metric space (X,d) , we talk about subsets of X being bounded iff ...
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Generalisation of posets

The notion of (set) monoid can be generalised to that of monoid object in a monoidal category. Can the notion of poset be generalised in a similar fashion? How?
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Recognizing linear orders embeddable in $\Bbb R^2$ ordered lexicographically

Let $(C, \leq_C)$ be any chain: binary relation $\leq_C$ on $C$ is reflexive, antisymmetric, transitive, and total. Give $\mathbb{R}^2$ the lexicographic order: for all real numbers $a,b,c,d$, $(a,b) ...
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Ascending Chain Condition

In the Adams - Loustaunau book about Gröbner bases, I found in the definition of noetherian ring that a ring satisfies the following condition is a noetherian ring: If $I_1\subseteq I_2\subseteq ...
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Showing that poset of set of supports of a vector space is semimodular

Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. The support of a vector $v = (v_1,\ldots, v_n) \in \mathbb{K}^n$ is given by ...
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What would be the join and meet of this lattice?

I'm working on the following problem: Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. Let $L$ denote the set of supports of all vectors ...
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What is reverse inclusion?

I'm learning about posets for the first time. What does it mean for a collection of sets to be "ordered by reverse inclusion"? Thank you.
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Topology and Monotone Convergence Theorem

I'm looking on the Monotone Convergence Theorem and asking myself whether it is the property of ANY order topology induced by some total order that is dedekind complete ( basis are the open-intervals ...
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Find if relation is partial / total ordered

These are two problems of the Velleman's How to prove book in which it has been asked to find if $R$ is an partial order or/and an total order: $R1 = \{(x,y) \in A ...
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Nested Interval Property and Ordered Fields

I have been given the following as a general version of the Nested Interval Property, which is a predicate whose free-variable F ranges through the universe of all ordered fields : Every ...
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Order and Metric

Consider the Reals as a totally ordered set via its natural order ( linear continuum ). Such order induces an order topology ( basis is the collection of open-intervals of that order), which ...
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Drawing a Hasse Diagram

I am trying to draw a Hasse diagram and wanted to see if anyone can let me know if I am doing it right. Let R = {(a,b) | a divides b} be a relation over the set {1, 2, 3, 4, 5, 12} That is what I ...
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Fact related to non-paracompactness in well-ordered spaces

Let $\omega_1$ be the first uncountable ordinal. Look at the set $X=[0,\omega_1[$, the set of all ordinals $<\omega_1$. It is a well-ordered set with a smallest element. Since any ordinal strictly ...
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Order Type of Positive Reals.

Let (M,d) be a metric space , $\varepsilon>0$ be a real number . Fix $p \in M$ and denote $U_\varepsilon(p) $ = {$x \in M$ : $ d(p,x) < \varepsilon$ }. Now define the order $\leq$ on the ...
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Order and Topology

Given any set, a total order on that set can induce a topology on that same set. Does the opposite also work ? Given a topology on a set, can it induce an order ( perharps total ) on that set ? ...
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How do the interval and lower topologies relate when generated by an arbitrary poset?

At first glance, it would seem that the lower topology (and the upper topology, for that matter) would be a subset of the interval topology for a partially ordered set P, since the open-ended ...
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Finding smallest set of real numbers according to a property

This is one of the example problem that has been solved in Velleman's How to prove book: Find the smallest set of real numbers X such that $5 \in X$ and for all real numbers $x$ and $y$, if $x ...
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Binomial-like expansion for non-commuting operators

I would like to evaluate the following and express it in a compact form: $(\hat{a}^\dagger(x)+\hat{a}(x))^n\,\vert0\rangle$, where the $n^{\text{th}}$ power of the sum of the annihilation and creation ...
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Understanding Dilworth's theorem

One formulation of Dilworth's theorem(for finite partially ordered sets) states that : There exists an antichain A, and a partition of the order into a family P of chains, such that the number of ...
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Construct an sequence in order set

Assume that we have an uncountable ordered set $\Theta$. Can we construct a sequence $(x_n)$ which $x_1<x_2<...<x_n<...$. In addition, for each $x\in \Theta$, there exists $n$ such that ...
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Can we define Möbius functions on preordered sets?

(From Rota's paper): Let $P$ be a locally finite poset. The incidence algebra of $P$ is the set of all real-valued functions $f$ of two variables $x$ and $y$ in $P$ with the property that $f(x, y) = ...
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Conditions for monotone function to take maximal chains to maximal chains srujectively

Suppose that $P$ and $Q$ are graded (with rank function $r$) connected posets with least elements and suppose that all maximal chains of $P$ and $Q$ have length $n$. Let $f:P \to Q$ be a surjective ...
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Help with the definition of ordinal sum

"Given two ordinals $\alpha$ and $\beta$, let $A = (\alpha$ x {0}) $\cup$ $(\beta$ x {1}). Then, define a well ordering on A by: $(v, i) <_{A} (\tau, j) \iff (i \lt j) \lor (i = j$ $\land$ $v ...
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show that N(G) is a lattice?

if $G$ is a group let N(G) be the set of normal subgroups of $G$ define $\wedge$ and $\vee$ on$ N(G)$ by $N_{1}\wedge N_{2}=N_{1}\cap N_{2}$ and $N_{1}\vee N_{2}=N_{1} N_{2}=\{n_{1}n_{2}:n_{1}\in ...
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Natural Ordering of the Class of Hermitian Preserving Maps

I am trying to understand Man-Duen Choi's remark 3 in his paper Completely Positive Linear Maps on Complex Matrices: For a linear map $\Phi : \mathcal{M}_{n} \to \mathcal{M}_{m}$, it is obvious that ...
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Strengthening of a theorem about filters vs a counter-example

Let $S$ be a non-empty set of filters on a meet-semilattice. If our semilattice is a distributive lattice, then the supremum (on the poset of filters ordered by set-theoretic inclusion) of $S$ is the ...
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$f\colon \mathbb R\to\mathbb R$ Knaster-Tarski Theorem

Knaster-Tarski Theorem says that for a function $f\colon L\to L$, where $L$ is a complete lattice, and $f$ is a increasing function, then a $f$ has a fixed point. Question: For a function $f\colon ...
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A question on order theory (an ordered set and its subset)

Filtrator is a pair $(\mathfrak A; \mathfrak Z)$ of a poset $\mathfrak A$ and it subset $\mathfrak Z$. Let $a\in\mathfrak A$. Then by definition $\operatorname{up} a = \{x\in\mathfrak Z \mid x\ge a ...
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equivalence of ordering properties

The wikipedia page on weak orderings states that Transitivity of incomparability (together with transitivity) can also be stated in the following forms: If x < y, then for all z, either x ...
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What is $|\emptyset|$? [closed]

What is $|\emptyset|$? Intuitively I would say $0$, but the number of partitions of $0$ is $1$, which is established due to the elegance of the generating function for partition numbers.
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Is A∩B has Maximum?

Consider $A=(0,1)∪\{2\}$ and $B=(0,1)∪\{3\}$. Both sets have maxima, $2$ and $3$ respectively. But $A∩B=(0,1)$ which has no maximum. How can we say that $A∩B=(0,1)$ does not have any maximum. ...