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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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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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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47 views

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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33 views

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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58 views

Is this a valid example of a poset?

I am trying to understand better the basic concepts of partially ordered sets, and tried my own examples by using modular-arithmetic. For this, I tried to define an algorithm to order integers, here ...
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20 views

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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38 views

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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31 views

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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40 views

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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39 views

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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23 views

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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34 views

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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25 views

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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38 views

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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28 views

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. ...
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Stochastic dominance characterization

Consider two probability measures on $\Bbb R$ given by $\mu$ and $\nu$. We write $\mu\leq \nu$ if there exists a joint distribution $P$ with the latter marginals such that $P(x\leq y) = 1$. In ...
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Bijection on Preordered Sets Implies Homeomorphism

Prove that if $X$ and $Y$ are finite, then the "converse" of one of my other questions Homeomorphism on a Preordered Set is true: if $h: X \to Y$ is bijective and satisfies $\forall a,b \in X, ...
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Proof that $S$ is a partial order when it is the reflexive closure of a strict partial order

Suppose $R$ is a strict partial order on $A$. Let $S$ be the reflexive closure of $R$. Show that $S$ is a partial order on A. ("How to Prove it", Chapter 4.5 exercise 4.a) To prove that $S$ is a ...
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Does the empty set have a minimum if it is a subset of R?

True or False: As a subset of $\bf{R}$: $\emptyset$ has a minimum. What is the difference between a supremum and the maximum? are they used interchangeably?
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What is subset partial order

This is one of the example problem 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 $x \in X$ and $x ...
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1answer
29 views

Number of directed acyclic graphs with specific topological sort

Given a topological sorting $>$ over a set of elements $N$, how many possible DAGs with vertex set $N$ that are consistent with $>$ ? ( note that a DAG may be consistent with other orders in ...
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22 views

A Directed Preorder from a Local Base

Prove that if $\mathscr{A}$ is a local base at point $p \in (X, \mathscr{T})$, then $\supset$ (the superset relation) is directed on $\mathscr{A}$. Here are the definitions I am given for this ...
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Supremum And infimum True or false

If the sets A and B have maxima and A ∩ B $\neq$ 0 , then A ∩ B has a maximum. Is this statement is true or false ? How to do these kind of problems
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Homeomorphism on a Preordered Set

Prove that if $h$ is a homeomorphism from $(X,\mathscr{S})$ to $(Y,\mathscr{T})$, then $\forall a, b \in X \left( a \trianglelefteq_{\mathscr{S}} b \iff h(a) \trianglelefteq_{\mathscr{T}} h(b) ...
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Are all finitely distributive and join-complete lattices infinitely distributive?

The infinite distributive law on a join-complete lattice $L$ is as follows: $\displaystyle a \wedge\left( \bigvee_{b \in B} b \right) = \bigvee_{b \in B}(a \wedge b) $ for all $a \in L$ and $B ...
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Duplicated Question in order theory [duplicate]

(Duplicate) I'm solving some questions in discrete mathematics, and I stuck with this problem. Prove that there are uncountably many non-isomorphic linear orderings in N.
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2answers
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Uncountably many ordering of natural numbers [duplicate]

How do I construct uncountably many non isomorphic ordering of natural numbers?
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Uniqueness of Topology on Finite Set

Prove that if $X$ is a finite set with preorder $R$, then exactly one topology $\mathcal{T}$ on $X$ satisfies $\trianglelefteq_{\mathcal{T}} =R$. > Here is the definition the mentioned preorder: ...
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Existence of a Embedding between two Posets

Prove that not every finite Partially ordered set can be embedded into $\Bbb R^3$ (ordinary $3$D space) with the partial ordering as follows. $(x_1,y_1,z_1)\preceq(x_2,y_2,z_2)$ if and only if ...
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Embedding between Two posets

Suppose $X$ is a finite poset (partially ordered set). Then does an embedding $f$ always exist between the finite poset $X$ and the $\Bbb R^3$ (ordinary $3$-d space)?
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Reflexivity and Order

It seems to me that the most important concept that the idea of an order brings, is tied to the notion of assymetry ( or the weak antisymmetry, if you may ). However, the way the order ...
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Preorder on Topological Space

Given a topological space $(X, \mathcal{T})$, define $\trianglelefteq_{\mathcal{T}} = \{(a,b) \in X^2 : a \in \overline{\{b\}}\}$. Prove that $\trianglelefteq_{\mathcal{T}}$ is a preorder, that is, ...
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Subbase for a topology defined by a family of closed sets?

Let $X$ be a set and $\mathcal{T}$ the smallest/weakest/coarsest topology containing some family $\left\lbrace C_i\right\rbrace_{i\in I} $ of sets declared to be closed. This is well-defined since ...
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Finding an uncountable chain of subsets the integers

How to find an uncountable subset of $P(\mathbb{N})$ such that every two elements of it can be compared. In fact, give an uncountable subset of $P(\mathbb{N})$ such that has totality property. We mean ...
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Dimension of a LOTS in $\mathbb{R}^2$

Equip $\mathbb{R}^2$ with the lexicographic order $(x_1,y_1) < (x_2,y_2)$ iff $x_1 < x_2$ or $x_1 = x_2 \Rightarrow y_1 < y_2$. Consider the induced order topology generated by the subbase ...