# 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 algebraic structures.

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### Characterising $\sigma$-algebras as posets

A $\sigma$-algebra is defined as a set $X$ together with a subset $\Sigma$ of the power set $\mathcal{P}(X)$, such that $X\in \Sigma$ $\Sigma$ is closed under complementation $\Sigma$ is closed under ...
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### What concept of order is introduced in the twentyfold way?

Four of the folds not present in the twelvefold way but introduced in the twentyfold way, rows $5$ and $6$ of the linked table, are defined by the statement that order matters. However, my ...
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### A proof of well-ordering theorem in “Set Theory and General Topology” by Fuichi Uchida.

I am reading "Set Theory and General Topology" by Fuichi Uchida. In this book there is the following theorem: Let $X$ be any set. There exists an order $\rho$ such that $(X, \rho)$ is a ...
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### Is every Hasse diagram connected?

Is every Hasse diagram connected? In other words - is some antichain subset of a poset, a poset in its own right? I ask because I see no connectedness condition in the definition, but never seem to ...
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### Non-emptyness of projective limit over countable set

I am doing exercise 1.9 from Lenstra's Galois theory for Schemes. Let $\left((S_i)_{i\in I},I,f_{ij}\right)$ be a projective system, where $I$ is countable, all $S_i$ are non-empty and all $f_{ij}$ ...
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### Chain that doesn't contain maximal element?

I cannot find a proof or example that every chain contains a maximal element (from Friedberg et al). A collection of sets is called a chain if for each pair of sets A and B in the chain, A ⊆ B or B ⊆ ...
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### Hausdorff Maximal Principle implies Well-Ordering Theorem

While looking at a proof of "Maximum Principle implies Well-Ordering Theorem" in this web-page https://proofwiki.org/wiki/Hausdorff_Maximal_Principle_implies_Well-Ordering_Theorem#:~:text=By%20the%...
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### If $X_i$ has supremum for any $i=1,…,n$ then $\sup\bigcup X_i=\max\{\sup X_i\}$.

Statement If $X$ is a totally ordered set and if $\mathfrak{X}=\{X_i\subseteq X:i=1,...,n\}$ is a finite subcollection of not empty subset of $X$ with supremum then $\bigcup\mathfrak{X}$ is limited ...
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### Hessenberg power of ordinals (redux)

This is a follow-up to this question, in which the given definition failed. Let $f : \varepsilon_0 \rightarrow \mathbb{N} \rightarrow \mathbb{N}$ recursively defined by \begin{align} f\left(\sum_{...
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### Prove that for k, the number of elements with prime order p, k = -1 (mod p)

Let p be a prime number and let G be a finite group whose order is divisible by p. Let k be the number of elements $x \in G$ of order p and let $l$ be the number of subgroups $H \subseteq G$ of ...
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### Partially ordered set with mixed-integer variables

I have a finite set of vectors $\mathcal{S} \subset \mathbb{R}^n$ with mixed-integer components (let's say $n_c$ and $n_i$, with $n = n_c + n_i$). I was wondering whether $\mathcal{S}$ is always ...
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### Given a boolean matrix $M$ what are the matrices formed by replacing $1s$ in $M$ with $0s$ called?

Given two boolean matrices $A$ and $B$ over some common dimensions one can form an order via $A\leq B\iff \forall i,j(A_{i,j}\leq B_{i,j})$ under this order, what would the matrix $A$ be called in ...
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### How can we express that a partial order is more complete than another one?

Suppose we have two partial orders $R$ and $I$ on $\mathbb{C}$ (conplex numbers) such that: $R$ is a total order on $\mathbb{R}$ (real numbers). $I$ is also a total order on $\mathbb{R}$ and, ...
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### Cauchy completion of transfinite “rationals”

Let the Hessenberg power $\alpha^\beta$ be the supremum of ordinals that are order-isomorphic to some well-order on the set of finite-support functions $\beta \rightarrow \alpha$ that extends the ...
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### Can a non-total order define a prepositive cone in a field?

Fact: A total order $\le$ that satisfies the ordered field axioms defines a set $\{x:0\le x\}=P$ that is a prepositive cone. I proved this. I had to invoke trichotomy (as well as other lemmas) to ...
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### Use a reflexive and transitive closure to transform an antisymmetric and acyclic relation into a partially ordered set.

The relation $R=\{(x,S_1),(S_1,S_2),(S_2,S_3),(S_3,S_4),(S_4,y)\}$ is antisymmetric and acyclic but not transitive or reflexive. We know that any antisymmetric and acyclic relation can be turned into ...
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### well founded induction / difference to strong induction

We are given a chocolate bar with $n$ pieces (squares) and we already know by strong induction that $n-1$ are needed to break it in individual parts. https://web.stanford.edu/class/archive/cs/cs103/...
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### dominating antichain

Let $P$ be a finite set with partial orderings $\leq_1$ and $\leq_2$. Then, there is a so-called dominating antichain $A \subseteq P$ where no two distinct elements of $A$ are comparabale in $\leq_1$ ...
### Find a binomial poset with factorial function $B(n) = q^\binom{n}{2}$
In Enumerative Combinatorics, Volume I, second edition,Example 3.18.3 e, page 323, Stanley describes this poset: Let $V$ be an infinite vertex set, fix $q \in \mathbb{P}$, and let $P$ be the set of ...