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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How can division on Natural numbers be a poset?

For natural numbers including 0, how can it be a poset while zero cannot divide zero. Doesn't this mean it isn't reflexive?
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Is Least Upper Bound of Empty Set equal to Greatest Lower Bound of a another Set?

We had this discussion today that $\operatorname{LUB}(\varnothing) = \operatorname{GLB}(L)$ in a complete lattice $(L,\leq)$. I'm not still not getting it why that is the case.
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partial DAG and number of linear orders

Let $>$ be a linear order relation over a set $A$. Consider the graph $G$ that represent the transitive closure of $>$. Obviously $G$ is directed and acyclic. Given a set of edges ...
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Proving that $B_1$ and $B_2$ doesn't have maximal element

This is one of the problem I have been solving form Velleman's How to prove book: Suppose $R$ is a partial order on $A$, $B_1 \subseteq A$, $B_2 \subseteq A$, $\forall x \in B_1 \exists y \in B_2 ...
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number of permutations that have $i<j$ against the ones with $j>i$

Consider a set $A=[a_1,a_2,\dots,a_n]$ and its all possible permutations $P$. Select one permutation $\sigma=(\sigma(a_1),\sigma(a_2),\dots,\sigma(a_n)\ )\in P$ and consider a set of distinct pairs ...
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Mimimum and Minimal element of an ordered set

During study of ordered sets,I gone through 'minimal element' and 'minimum element' of a set. Can someone explain the difference between them with examples? Many thanks and regards,
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from linear extension to partial order

Going from partial order to a linear one is easy, we just do topological ordering. I am wondering about the other way around. In particular: let $>$ be a linear order over a set $A$. Let $\Gamma$ ...
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Isn't every totally ordered set well-ordered?

A well order is a total order on a set $S$ with the property that every non-empty subset of $S$ has a least element. But surely it follows from the definition of a total order that any non-empty ...
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Three theorems for the price of one? (like duality)

Why the notion of "duality" (when we get two theorems for the price of one) are ubiquitous in mathematics (order and lattice theory, category theory, group theory), but "triality" (three theorems for ...
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29 views

Order relations and counting the number of cases (Fubini numbers)

I have $n=2$ numbers $a$ and $b$, $a\in\Bbb{N}$ and $b\in\Bbb{N}$. Then I have the function $f$ defined as: $ f(x,y) = \begin{cases} -1, & \text{if $x<y$} \\ 0, & \text{if $x=y$} \\ +1, ...
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Why is the degree condition for a degree reverse lexicographic order necessary?

A degree reverse lexicographic order $\prec$ is defined as follows: Given the polynomial ring $R=K[x_1,...,x_n]$. Two monomials in $R$ have the order $x^u\prec x^v$, if $\deg(x^u)<\deg(x^v)$, or ...
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Partial ordering of functions

Let $X$ be the set of all real-valued functions $x$ on the interval $[0,1]$ and let $x \leq y$ mean that $x(t) \leq y(t)$ for all $t \in [0,1]$. Does it define a partial ordering/ total ordering? Does ...
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1answer
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Is a lexicographic order over a partial-order and a total-order a total ordering?

Is the following statement true? If so, can someone advise on how to prove it? Let A = $(S_1,\prec_1)$ be a partially-ordered set and let B = $(S_1,\prec_2)$ be a totally-ordered set Let ...
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2answers
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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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1answer
36 views

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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1answer
28 views

$p$-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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1answer
35 views

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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24 views

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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1answer
26 views

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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1answer
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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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1answer
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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 ...