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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a problem in Stein's book 'Real analysis', relate to continuum hypothesis.

The question is from chapter 2, problem 5 in Stein's book 'Real analysis': 5.There is an ordering $≺$ of $\mathbb R$ with the property that for each $y\in\mathbb R$ the set $\{x\in\mathbb R : x ≺ ...
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Lexicographic order on $\alpha^\beta$ is well-ordered

Suppose that $\alpha$ and $\beta$ are ordinals, and define the following order on $\alpha^\beta$, the set of all functions $\beta\to\alpha$ with finite support: $$f\,R\,g\iff\exists ...
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+50

Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
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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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What do we call well-founded posets whose elements have a unique height?

What do we call those well-founded posets $P$ with the property that for every $x \in P$, all maximal chains in the lowerset generated by $x$ have the same length? Examples: The set of all finite ...
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35 views

Zorn's lemma converse? (Context: Maximal proper subgroups)

So, in my qual prep class a pretty simple question popped up: "Prove that for any nontrivial finite group there exists a maximal proper subgroup." So of course, my natural inclination was to ...
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Order by inclusion

Given the set $E = \{a,b,c\}$, order the power set of $E$ (all subsets) by inclusion. I think the order would be $\varnothing$, $\{a\}$, $\{b\}$, $\{c\}$, $\{a,b\}$, $\{a,c\}$, $\{b,c\}$, ...
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$|\alpha\times\alpha|=|\alpha|$ for infinite ordinal $\alpha$, definably

I am trying to prove Proposition 1.7 of http://math.bu.edu/people/aki/7.pdf: Proposition 1.7 (Halbeisen–Shelah). If $\aleph_0\le|X|$, then $|{\cal P}(X)|\not\le|{\rm Seq}(X)|$. In words, ...
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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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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 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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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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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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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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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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Finitely generated ordered monoids and noetherian subsets

Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order defined by "$x\leq y$ if and only if there exists $z\in E$ with $y=x+z$", so ...
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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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Total Order Relation

Given $X = \{ a , b , c \}$ and $R = \{ (a,a) , (b,b) , (c,c) , (a,b) , (a,c)\}$ Is $R$ a total order on $X$? I know that total order requires the relation to be comparable on all elements, ...
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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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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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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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Describing the order in symmetric group.

Once again I got stuck in my work. I'm not sure how to understand these conditions, let alone trying to figure out the proof. Any explanation will be greatly appreciated. Next we define a partial ...
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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 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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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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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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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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Is any set that has a preorder is a directed set?

In the definition of directed set, they emphasized that aside from it having a preorder, "every pair of elements has an upper bound." My question is that isn't the latter property implied by the ...
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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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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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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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Recommendation for Order Theory texts

Order Theory, Lattice Theory, or any directly relatable subject is not taught at my university, and quick searches don't give much clue into good textbooks for the subjects. My question is: what are ...
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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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Counting permutations that respect a partial order

Suppose you have three kinds of coins, say: pennies, nickels, and dimes. Each penny has a unique date, likewise for nickels and dimes. (A penny and a nickel may have the same date, etc.) How many ways ...
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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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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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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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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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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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If a partial order has just one unique maximal element, must that element be a greatest element of x?

I believe that it does not have to be a greatest element, because to be a greatest element it must be comparable to all the x in X. I just don't know how to go about proving this.
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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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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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$p$-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 ...