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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An order type $\tau$ equal to its power $\tau^n, n>2$

In this question we are concerned only with linear (aka total) order types. By a cardinality of an order type we understand a cardinality of an instance of this type, which obviously does not depend ...
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$<$ on a preorder is a strict partial order

Definition: Suppose $X$ is a preorder. Define $x < y$ as $x \le y$ and $y \not\le x$ for each $x, y \in X$. Question: Show that this gives a strict partial order on $X$.
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Uniqueness of meets and joins in posets

Exercise 1.2.8 (Part 2), p.8, from Categories for Types by Roy L. Crole. Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
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238 views

Examples of preorders in which meets and joins do not exist

Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
10
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1answer
373 views

Linearly ordered sets “somewhat similar” to $\mathbb{Q}$

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{\eta} + \pmb{1}$ - order type of $\mathbb{Q}\cap(0, 1]$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. Let's say that a linear ...
16
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2answers
343 views

Is $(\pmb{1} + \pmb{\eta})\cdot\pmb{\omega_1} = \pmb{1} + \pmb{\eta}\cdot\pmb{\omega_1}$?

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{1}$ - order type of a singleton set. $\pmb{\omega_0}$ - order type of $\mathbb{N}$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. ...
10
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1answer
170 views

Is $\pmb{\eta}\cdot\pmb{\omega_1} = (\pmb{\eta} + \pmb{1})\cdot\pmb{\omega_1}$?

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{1}$ - order type of a singleton set. $\pmb{\omega_0}$ - order type of $\mathbb{N}$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. ...
6
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1answer
138 views

Poset of idempotents

Let $A$ be an associative algebra and consider the poset $I$ of idempotents in $A$, where as usual if $e,f \in A$ we say $e \leq f$ provided that $e = fef$. Clearly $0 \in I$ is minimal, but in ...
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1answer
133 views

Does an isomorphism induce an order isomorphism?

Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a \curlyvee b$ if only if there is a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$. Let ...
4
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2answers
159 views

Is every order isomorphism of sets induced by a bijection?

Let $f$ is an order isomorphism $\mathscr{P}A \rightarrow \mathscr{P}B$ (where $A$ and $B$ are some sets, the order is set-inclusion $\subseteq$). Is it true that it always exist a bijection $F: A ...
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1answer
120 views

Are homeomorphisms order-isomorphisms?

Is every homeomorphism between topological spaces an order isomorphism (for orders of inclusion $\subseteq$ of sets)?
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223 views

Set of maximal chains in $\langle\mathcal{P}(\mathbb{N}),\subseteq\rangle$

Let $S$ be the set of all maximal chains in the poset $\langle$$\mathcal{P}$$($$\mathbb{N}$$),$$\subseteq$$\rangle$ partitioned into equivalence classes by their order types. How many different order ...
3
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1answer
182 views

Down-sets in posets and directed sets

Let P be a poset and let us say that a subset A of P is a down-set if: $$x \in A, y < x \implies y \in A.$$ A directed set is a poset P such that for every two elements, $a,b \in P$ we can find ...
2
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1answer
223 views

Preorders, chains, cartesian products, and lexicographical order

Definitions: A preorder on a set $X$ is a binary relation $\leq$ on $X$ which is reflexive and transitive. A preordered set $(X, \leq)$ is a set equipped with a preorder.... Where confusion cannot ...
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2answers
142 views

$<$ in a preorder

The author of the book I am studying defines $<$ for a poset as If $x, y \in X$, where $X$ is a poset, then we shall write $x < y$ to mean that $x \le y$ and $x \ne y$. From this, I can ...
2
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1answer
366 views

Does every set have a well ordering with greatest element?

Does every non-empty set have a well ordering with greatest element? It is well known that every set has a well ordering. But can we also assume that this well ordering has greatest element? [Edited ...
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1answer
126 views

Is the Kleene/Brouwer ordering dense?

This question was motivated by a statement in Simpson's Subsystems of Second Order Arithmetic (second edition), p. 168. It is straightforward to verify (in $\mathsf{RCA}_0$ for instance) that ...
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3answers
112 views

Equal elements vs isomorphic elements in a preoder

While introducing preorders, Roy L. Crole in Cateories for Types states If $x \leq y$ and $y \leq x$ then we shall write $x \cong y$ and say that $x$ and $y$ are isomorphic elements. I'm trying ...
2
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0answers
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Ordering of multisets in “Paramodulation based theorem proving”

I'm reading this paper: http://www.lsi.upc.edu/~albert/papers/handbook.ps.gz and I can't understand a part of it. it defines an ordering on multisets (it defines a multiset over $A$ as a function $A ...
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1answer
71 views

From an equality to a comparison

Let $X$ be a set. Let $0$ be an element of $X$. For a function $P$ defined on tuples of $n$ elements of the set $X$ we know (for every tuples $f$ and $g$ each having $n$ elements) $$\forall i \in n : ...
6
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2answers
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partially ordered group, positive cone, quotient (exercise)

Definitions: A partially ordered group or po-group is a po-set $(G,\leq)$, such that $G$ is a group and $\forall x,y,a,b\!\in\!G\!:x\!\leq\!y\Rightarrow axb\!\leq\!ayb$, i.e. a po-set that is a group ...
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1answer
234 views

Are lexicographic and multiset the only extensions of sets that preserve well-foundedness?

It is well known that for a given set $S$ with well-founded order relation $R$, the lexicographic order that extends $R$ on tuples of $S$ is also well-founded. Also, the multiset order on the ...
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2answers
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For order-reversing maps between posets/lattices, is the inverse always order-reversing?

I am having trouble with the following lemma from J. Rotman's Galois Theory book: Lemma $83$. If $L$ and $L'$ are lattices and $\gamma: L \rightarrow L'$ is an order reversing bijection $[a \leq ...
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1answer
352 views

Lexicographical order - posets vs preorders

I found the following definition for lexicographical ordering on Wikipedia (and similar definitions in other places): Given two partially ordered sets $A$ and $B$, the lexicographical order on the ...
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1answer
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Question of an isomorphism of $\epsilon_ 0$ and a subset of the rationals.

I don't know if this question is appropriated for this site. Anyway, I'm searching for an isomorphism of order $f:K \longrightarrow \epsilon_o $, such that $(K, \leq)$ is a subset(proper or not) of ...
3
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1answer
62 views

Round a value in a set of number

Given an order set of some number, $S=\{1.3, 1.7, 1.9, 2.8\}$, I would like to know how can I mathematically define a function that round a value to the nearest number in the set $S$. For example, if ...
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2answers
192 views

A sufficient condition for order isomorphism of posets?

Let $\mathfrak{A}$ be a poset. For $a, b \in \mathfrak{A}$ we will denote $a \not\asymp b$ if only if there are a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$. Let ...
3
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1answer
167 views

Do filters on a Boolean algebra also make a Boolean algebra?

Let $\mathfrak{B}=(B,\bot,\top,\lnot,\wedge,\vee)$ be a boolean algebra. $B_F$ be the set of all filters on $\mathfrak B$. And for all filter $F$, $G$, $F \wedge_{B_F} G \colon= \mathbf C(F \cup G)$ ...
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1answer
369 views

MacNeille completion of a totally ordered set: Dedekind cuts

If $X$ is any partially ordered set with $A\!\subseteq\!X$ and $x\!\in\!X$, define $x\!\leq\!A :\Leftrightarrow \forall a\!\in\!A\!: x\!\leq\!a$ and $A\!\leq\!x :\Leftrightarrow \forall a\!\in\!A\!: ...
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2answers
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Can such a function exist?

Denote by $\Sigma$ the collection of all $(S, \succeq)$ wher $S \subset \mathbb{R}$ is compact and $\succeq$ is an arbitrary total order on $S$. Does there exist a function $f: \mathbb{R} \to ...
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84 views

Name for a discrete ordered set with finite subranges

Is there a term for a set that is: discrete totally ordered has finite subranges (not a technical term), i.e. for any a and b, $\{x|a<x<b\}$ is finite At first glance, it seems this implies ...
3
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1answer
363 views

Cardinality of the set of ultrafilters on an infinite Boolean algebra

Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?
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5answers
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Are there methods to well order a finite group in a meaningful way?

Can some finite groups be well ordered in a "meaningful" way? I mean, it is clear that we can trivially find a bijection between $\{1,...,n\}$ and a finite group $G$ with $n$ elements, but I am ...
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1answer
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Inducing maps between Boolean completions of posets

Given a separative poset $P$, we can form it's Boolean completion $B(P)$. This is a Boolean algebra whose elements are regular cuts on $P$, as defined here. Also, $P$ embeds densely into $B(P)$ by ...
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Stone duality for ideals and filters (exercise)

In A Course in Universal Algebra (Burris, Sankapannavar), the exercise 4.4.7-8, p.158, says: Let $A$ be a Boolean algebra. Denote $A^\ast:=\{\text{ultrafilters of }A\}$, and give $A^\ast$ the ...
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Are the Join and Meet operators on complete lattices both continuous?

Suppose that $(A,\le)$ is a complete lattice, that means $(A,\wedge,\vee)$ is a lattice which satisfies $$\forall B \subseteq A[\bigwedge B\text{ and }\bigvee B\text{ exist}].$$ And of course ...
9
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3answers
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A “Cantor-Schroder-Bernstein” theorem for partially-ordered-sets

If A and B are partially-ordered-sets, such that there are injective order-preserving maps from A to B and from B to A, is there necessarily an order-preserving bijection between A and B ?
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1answer
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$M_3$ is a simple lattice

I'd like to prove (exercise 9.5 in Roman's Lattices and Ordered Sets, p.203) that the lattice $M_3$ is simple, meaning that the only congruences on $M_3$ are the trivial ones (the 'equality' ...
2
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1answer
138 views

ideals of a ring form a modular lattice

We know that if $M$ is a left $R$-module, then $(\{\text{submodules of }M\},\subseteq)$ is a modular lattice. Taking $M\!=\!R$, we deduce that $(\{\text{ideals of }R\},\subseteq)$ is a modular ...
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Isomorphisms: preserve structure, operation, or order?

Everyone always says that isomorphisms preserve structure... but given the (multiple) definitions of isomorphism, I fail to see how the definitions equate with the intuitive meaning, which is that two ...
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1answer
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When is an interval (order theory) = line segment (in $\mathbb{R}^n$)?

If $(X,\leq)$ is any poset and $a,b\!\in\!X$, then we define the interval as $$[a,b]_\leq:=\{x\!\in\!X;\;a\!\leq\!x\!\leq\!b\}.$$ If $a,b\!\in\!\mathbb{R}^n$, then we define the line segment as ...
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Minimum set of component wise ordering relation

$\leq_{cw}$ is the partial component wise ordering relation. Given a set $M \subset \mathbb{N}_0^n$ we define $N = \min_{cw}(M)$ with $v \in N$ iff if $u \in M$ and $u \leq_{cw} v$ then $u = v$. I ...
2
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1answer
202 views

Number of possible partial orderings on a finite set

I've been reading about lattices and partial orders (my reference: Applied Abstract Algebra by Lidl, Pilz) while this question struck me. Let X be a finite set. Is there any way to determine the ...
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3answers
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Example of directed set (directed towards $x_0$)

On this page there are some examples of directed sets. One of those cites: "If $x_0$ is a real number, we can turn the set $\mathbb{R} − \{x_0\}$ into a directed set by writing $a \leq b$ if and ...
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411 views

Example of total order with some properties that is not well ordered

Is there an example of a total order with properties there is a least element and every element has a (unique) successor not is not also a well ordering?
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1answer
114 views

The Real line uniqueness …

I have a question in the following exercise: Let $\langle X, <\rangle$ be a total ordering with no first or last element, connected in the order topology and separable.Show that $\langle ...
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Examples of proofs by induction with respect to relations that are not strict total orders.

I have read this Wikipedia article and found it fascinating. I came across it after I tried to prove a certain statement with a method resembling induction in the set of natural numbers but ordered by ...
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1answer
130 views

“unexpected” isomorphism between finite posets?

The set of all divisors of a square-free number, partially ordered by divisibility, is trivially isomorphic to the set of all subsets of the set of prime factors, partially ordered by inclusion. Are ...
2
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1answer
234 views

algebraic closure and real closure are closure operators?

Are the algebraic closure (of a field) and the real closure (of a totally ordered field) closure operators (when restricted to appropriate sets of fields, so that they are maps on a set instead of a ...
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438 views

Is a chain-complete lattice a complete lattice without the axiom of choice?

This question is inspired by this question. Consider the following result: Let $(L,\leq)$ be a chain-complete lattice. Then $(L,\leq)$ is a complete lattice. Can this result be proven without ...