# Questions tagged [order-theory]

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. Order theory is not about the order of a group nor the order of an element of a group or other alegbraic structures.

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### Prove the inequality $breadth(P) \leq dim(P)$

Definition 1: Let n be a positive integer. We say that an order P has breadth at most n if for all elements $x_0$, $x_1$, ..., $x_n$, $y_0$, $y_1$, ..., $y_n$ in P, if $x_i \leq y_j$ for all $i \neq j$...
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### Partially ordered set's maximal and minimal

I have a few questions about a general partially ordered set which derive from a specific one. $R$ is a partially ordered set over $A=\{a,b,c,d\}$ (4 different item group). $S=R\cup {(a,b)}$ is an ...
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### Check my understanding of proof that Partial order can be extended to total order

Induction step : let n be arbitrary natural number, and suppose that every partial order on a set with n elements can be extended to total order. Suppose A has n+1 elements and R is a partial order on ...
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### Zeta polynomial of the Boolean Lattice

The task at hand is to compute the zeta polynomial $Z(B_k, n)$ of the Boolean Lattice in $k$ elements, which is the lattice formed by the subsets of $\left\{1, \dots, k\right\}$ under inclusion. The ...
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### Well ordered set in both directions is finite

So I tried to prove the following statement: If $(X, \leq$) and $(X, \geq)$ are well orders, then X is finite. But I'm not sure wether it's entirely correct. Proof: A well-ordered set has the ...
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### Multiplicative absolute value

The absolute value of a real number $r$ is defined to be the additive inverse of $r$ is $r < 0$, and $r$ is $r \geq 0$, where $0$ is the additive identity of the commutative group $\mathbb{R}$. ...
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### Cofinal sets and different definitions

Suppose that $A$ is an ordinal and $B\subset A$. We say that $B$ is Cofinal in $A$ iff $\sup(B) = A$. However, elsewhere I've read definitions that make no reference to ordinals. Here is the ...
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### Is there a difference between a range and an interval?

Can the terms 'interval' and 'range' be used interchangeably or do they describe different things? I am talking specifically about sets of numbers under a suitable $<$ relation, such that they can ...
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### Noetherianity of $\mathbb{R}^+$

I have some confusion about some properties of noetherian posets. I am defining a poset $X$ to be $\textbf{noetherian}$ if every ideal of $X$ Is finitely generated. I am trying to show that this is ...
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### How to prove the minimum identity for finite number of real numbers

I have trouble in proving the following statement: Let $n$ be an integer no less than 3, and $a_1,\dots, a_n$ be real numbers. If $n$ is even, then \begin{align*} &\min\{a_1,a_2,a_3\}+...
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### Which properties are invariant for totally ordered sets?

I am asked to find some number of totally ordered sets such that no one is isomorphic to a subset of another. To do this I wanted to think about the properties that are preserved for totally ordered ...
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### Understanding when a strict partial order can be satisfied by a set of integers

I am playing with a problem (that's part of a much bigger problem) and trying to understand the best way to approach it and also if there are existing names for this or similar problems. If I have a ...
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### Riesz space of functions from any set $X$ to $\mathbb{R}$

We collect certain functions from $f:X\to\mathbb{R}$ to $\mathscr{T}$ so that it is a vector space over $\mathbb{R}$ under the usual function addition, scalar product. Now we define an partial order ...
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### Lexicographical covering of boolean poset

Consider the boolean poset $2^{[n]}$. Is it true that for each rank $k< \frac{n}{2}$ and each positive integer $t\le \binom{n}{k}$, there is a perfect matching between the first $t$ elements of ...
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### Prob. 8 (b), Sec. 10, in Munkres' TOPOLOGY, 2nd ed: The union of any collection of disjoint well-ordered sets indexed by some well-ordered set …

Here is Prob. 8, Sec. 10, in the book Topology by James R. Munkres, 2nd edition: Problem 8 (a): Let $A_1$ and $A_2$ be disjoint sets, well-ordered by $<_1$ and $<_2$, respectively. Define ...
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### chain decomposition on partial order set

I am trying to do exercise in this book (page 54, number 3.11). The question : Find orthogonal chain decomposition as required in the proof of theorem from Shearer & Kleitman for $n=3$ and $n=4$. ...
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### Way of Ordering a ranked list

I am trying to find a good way of ordering and ranking a list of objects. They will be ranked multiple times by different people but i want their rank (out of 10) to update. e.g I rank the objects ...
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### Function $f:\mathbb{Z}^+\rightarrow\mathbb{Z}^-$ such that $m<n\implies f(m)<f(n)$?

The title says it all; I'm wondering if there is a sequence $\{f(n)\}_{n<\omega}$ of negative integers such that successive terms get strictly larger. The issue seems to be that as soon as we fix ...
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### Consider $A=^*B$ if and only if $A\triangle B$ is finite, What type of relation is $=^*$

Consider $A=^*B\iff A\triangle B$ is finite ($\triangle$ is the symmetric difference) what type of relation is $=^*$ First i start by checking if it is reflexive, anti-symmetric and transitive. so ...
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### Concerning the existence of the $R-$smallest element of a subset of a partially ordered set $A$

Is the following Proof Correct? Theorem. Given that $R$ is a total-order on $A$, every finite, non-empty set $B\subseteq A$ has an $R-$smallest element. Proof. We construct the proof by recourse to ...
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### Higher order version of Chebyshev's (algebraic) inequality

Let $(a_{i})_{i=1}^{n}$ and $(b_{i})_{i=1}^{n}$ be finite monotone increasing sequences (or both mon. decreasing) of real numbers and $m_{i}>0$ with $\sum_{i=1}^{n}m_{i}=1$. Chebyshev's (algebraic) ...
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### Extension of a continuous preorder

Given a compact metric space $X$ and a preorder $R$ on $X$ (a reflexive and transitive relation) which is continuous in the sense that the upper and lower contour set of any point, \begin{equation*} U(...
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### Poset test and Hasse diagram in GAP.

Given a finite set $X=\{x_1,...,x_r\}$ with a function $f: X \times X \rightarrow \mathbb{Z}$ on it, define a relation on it by $m \geq n$ iff $(m=n$ or $f(m,n)>0)$. Now for every element $m \in X$,...
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### Closure interpretation of a lattice

Hello I am doing a bit of reading on lattices right now and I would appreciate some help. A species of algebra with meet and join operations and binary relation which is quasi ordering can be ...
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### Find formula for number of dominated vectors in partial order

Below I consider vectors of same length $n$ and consisting of only $1$ and $-1$ elements. Let's call vector $v_1$ dominating for $v_2$ if count of $-1$ in $v_1$ is not less then in $v_2$ and position ...
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### Compactness of modal logic by proving existence of a model for a maximal set (using Zorn's lemma)

TASK: Given a set $\Gamma$ of propositional modal formulas where every finite subset of $\Gamma$ is satisfiable (say: $\Gamma$ is finitely satisfiable), show that $\Gamma$ itself is satisfiable. IDEA:...