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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Using substitution to make an equation into a separable differentiable equation

I have the question: By making the substitution $y = t^nz$ and making a cunning choice of n, show that the following equations can be reduced to separable equations and solve them. $$\dfrac{dy}{dt} = ...
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42 views

Proving that every submodel of DLOE is an elementary submodel

Let $A$ and $B$ be models of the theory of Dense Linear Orders without Endpoints such that $|B| \subset |A|$. I'm trying to prove that $B$ is an elementary submodel of $A$. Using Tarski-Vaught test I ...
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27 views

Does multiplication by the inverse of a Cholesky matrix preserve order?

Let $n \in \{1, 2, \dots\}$ and let $C \in \mathbb{R}_{n, n}$ be a real, symmetric and positive definite $n \times n$ matrix. Define $B \in \mathbb{R}_{n, n}$ to be the real, lower triangular matrix ...
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proving properties of (graph) dominance defined via a system of equations

Some notions on graphs can be defined via a system of equations with values in a lattice. For example, dominance $d(v_1, v_0)$ ($v_1$ dominates $v_0$) in a graph $g$ is defined by a system $\forall ...
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About elements of a poset

Consider a poset $\mathfrak{A}$. I denote $a\not\asymp b$ iff there is a non-least element $x\in\mathfrak{A}$ such that $x\le a$ and $x\le b$. I denote $\star a = \{ x\in\mathfrak{A} \mid x\not\asymp ...
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Prove that if a function f is continuous then it is monotonic

Let ($L_1 , \le_1 $) and ($L_2 , \le_2 $) be two posets. A function $f : L_1 \to L_2 $ is continuous if $ \forall E_1 \subseteq L_1, E_1 \neq \emptyset, f(LUB(E_1)) = LUB(f(E_1)) $ Prove that if $f$ ...
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Is this ordered square mathematically interesting?

Build a $5\times5$ (say) square using $\frac{1-\left(\frac{3}{2}\right)^n}{(2 m-1) 2^n-1}+1$, $$ \left( \begin{array}{ccccc} 0.5\hfill & 0.9\hfill & 0.944444 & 0.961538 & 0.970588 \\ ...
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Is possible write $\gamma \in [2, 0)$ under certain condition?

I am writing an academic paper. I have a variable that vary its value from $2$, when speed is zero, until $0$ at high speed. So, I would like write that $\gamma \in [2, 0)$ and not $\gamma \in (0, ...
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26 views

Proving $S_1 \subseteq S_2$ for transitive closure

This is one of the problem I have been working from Velleman's How to prove book: Suppose $R_1$ and $R_2$ are relations on $A$ and $R_1 \subset R_2$l (a) Let $S_1$ and $S_2$ be the reflexive ...
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26 views

simple clarification of equivalence relation and order relation notation meaning

So by definition an equivalence relation is a binary relation on a subset of the cartesian product of our set A, i.e $A\times A$ satisfying the 3 conditions. But I'm a little confused about ...
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10 views

On the matter of order completion (Dedekind completion)

So I was introduced recently to this idea of "order completion" of partially ordered set. A more special case is when this poset is in fact a completely ordered set, such as rationals. Completing the ...
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1answer
52 views

The least relation which produces a partial order

Fix some set $U$. Let $E$ be a binary relation on $U$. By definition $S(E)=\operatorname{id}_U \cup E \cup E^2 \cup E^3 \cup\dots$. I define $\mu(E) = \bigcap \{X\in\mathscr{P}(U\times U) \mid ...
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49 views

group of functions $\Bbb N \to \Bbb N$

$F$ is the group of all functions fron $\Bbb N$ to $\Bbb N$, $S$ is a relation on $F$: for $f,g\in F: (f,g)∈K \iff$ for all $n\in\Bbb N, f(n)≤g(n)$. So what is $g$ or $f$ ? Are they the outputs of ...
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Prove $a \leq b$, $\mbox{GLB}(\{a,b\}) = a$ , $\mbox{LUB}(\{a, b\}) = b$ are equivalent

It seems really obvious to be as a set containing $a$ and $b$ have GLB as $a$ and LUB as $b$. But how can it be proved mathematically?
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1answer
64 views

Prove divisibility is a partial order relation over natural numbers

I have proceeded like this so far: (this natural number set includes $0$) Reflexivity: Proof: $a = k \cdot a,$ since $k$ also belongs to natural number, it is proved. Anti-symmetry: $a \mid b$ ...
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31 views

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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24 views

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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1answer
19 views

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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69 views

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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37 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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37 views

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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36 views

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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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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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
41 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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$p$-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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111 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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1answer
74 views

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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25 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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29 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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33 views

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) ...