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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How to deal with Ideals generation from a Poset of sets including the empty set?

This question is strictly connected with this one: Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset. I started this question a week ago and I am still struggling on ...
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386 views

Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset

Theoretic background Lets just set some background to be sure what I am talking about Posets A Poset is defined to be a couple $(S,\leq)$ where a set $S$ and its elements $s \in S$ are connected ...
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346 views

Need construction for coequalizer in $\mathbf{Poset}$

My question can be stated quickly: I would like to see a construction of the coequalizer of two arbitrary Poset morphisms (along with a proof of its correctness, of course). Thanks! (The ...
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Simplest Example of a Poset that is not a Lattice

A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a ...
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111 views

Generated/induced/weak/strong partial orders

Let $f:X \to Y$ be a function. Suppose first that $\leq_X$ is a partial order on $X$. Is it possible to define a partial order $\leq_Y$ on $Y$, induced by $\leq_X$ (and optimal in some way), so ...
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Reference request: Indecomposable representations of posets

Let $I$ be a finite poset. Definition: A representation of $I$ is a functor $I\to\mathrm{Vect}_{\mathbb C}\ $. Equivalently, a representation of $I$ is a module over its incidence algebra $\mathbb C ...
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163 views

number of linear orders

It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved ...
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Total ordering of the reals with a certain property

I was communicated the following 1994 Miklos Schweitzer problem: Is there an ordering of the real numbers such that whenever $x<y<z$ (in this ordering), we have $y \neq (x+z)/2$? I really ...
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Construction of a partial order on a quotient of a coproduct

[NB: Throughout this post, let the subscript $i$ range over the set $\unicode{x1D7DA} \equiv \{0, 1\}$.] Let $(Y, \leqslant)$ be a poset, and $X\subseteq Y$. Let $\iota_i$ be the canonical ...
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390 views

Why do torsion-free abelian groups admit linear orders?

I have read a theorem that says that every torsion-free abelian group admits a linear order. The proof used tensor products and so was above my head. I tried to find another proof on the web and I ...
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Are irrational numbers order-isomorphic to real transcendental numbers?

I know that rational numbers are order-isomorphic to real algebraic numbers. Does it imply that irrational numbers are order-isomorphic to real transcendental numbers? I know that the order type of ...
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113 views

Section filter of a net and filter generated by a net

From Planetmath: Let $X$ be a set and $(x_i)_{i\in D}$ a non-empty net in $X$. For each $j\in D$, define $S(j):=\lbrace x_i\mid i\le j\rbrace$. Then the set $$S:=\lbrace S(j)\mid j\in ...
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2answers
178 views

The smallest filter (base) containing a subset of a power set

From Wikipedia's article for filter on a set: $P(S)$ is the power set of a set $S$. Given a subset $T$ of $P(S)$, we can ask whether there exists a smallest filter $F$ containing $T$. I was ...
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1answer
185 views

Cofinal subset and maximal elements in a poset

Let $(P;\leq)$ be a poset. A subset $A$ of $P$ is said to be cofinal in $P$ if for every $x$ in $P$ there is a $y$ in $A$ such that $x \leq y $. I was wondering if it is true that a subset of $P$ ...
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542 views

A question regarding the Continuum Hypothesis (Revised)

Given that the order type of the reals $(\mathbb R,<)$ has no definable points, can it be proven that $|\mathbb R|= \aleph_1$ by virtue of the fact that every uncountable proper subset $S$ of ...
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2answers
81 views

Does there exist an ordered field where only addition preserves positivity?

I am trying to find a field (I'll settle for other stuff - some type of ring) that is ordered and $a>0,\,b>0$ implies $a+b > 0$ yet $a,b > 0$ doesn't imply $ab > 0$. Much of the ...
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Is there an element with no fixed point and of infinite order in $\operatorname{Sym}(X)$ for $X$ infinite?

Let $X$ be an infinite set. Let $\operatorname {Sym}(X)$ denote the group of all bijections from $X$ onto itself. I have been thinking about the existence of elements of infinite order in this group. ...
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1answer
449 views

Rudin Theorem 1.11

After spending a few hours trying to understand Theorem 1.11 in Rudin's Principles of Mathematical Analysis, I still don't follow the proof. 1.11 Theorem Suppose $S$ is an ordered set with the ...
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153 views

Partial functions and fixed points

Let $\Phi : [\mathbb N \rightharpoondown \mathbb N] \to [\mathbb N \rightharpoondown \mathbb N]$ be the map from the set of partial functions $\mathbb N \to \mathbb N$ to itself (what's a nice way of ...
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93 views

Completeness of a poset

Suppose I have a poset $(P,\leq)$, and am trying to prove that it is complete. If, for a general subset $S \subseteq P$, I've come up with a candidate $x$ for its supremum and am trying to prove it is ...
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Question on order ideals

On p. 80 of his General topology, John Kelley gives the following theorem: Let $X$ be a distributive lattice, and let $A \subseteq X$ be an ideal and $B\subseteq X$ a filter such that $A\cap B = ...
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Examples of Galois connections?

On TWF week 201, J. Baez explains the basics of Galois theory, and say at the end : But here's the big secret: this has NOTHING TO DO WITH FIELDS! It works for ANY sort of mathematical gadget! If ...
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276 views

How to prove if an infinite set is total ordered?

Given the function: $$f:(a,b)\in\mathbb{Z}\times\mathbb{Z}\longrightarrow ab^2\in\mathbb{Z}$$ What can you say about injectivity and surjectivity? This is not injective. I have easily find ...
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Does the set $\mathbb{Z}$ satisfy the completeness axiom?

The Completeness Axiom states: A set $\mathcal{A}$ satisfies the Completeness Axiom if for every of its non-trivial and bounded subsets, a supremum exists in $\mathcal{A}$. If this is the ...
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A poset that's the union of the lower sets

Let $(P,\leq)$ be a poset, and let $\downarrow\! p = \{ x\leq p\}\subseteq P$. Let $M\subseteq P$ be the subset of all maximal elements of $P$. Question: is there a specific term for a poset ...
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About function inj, surj and something else. Is this exercise resolved correctly?

This is my problem: For every couple of integers $(a,b)\in\mathbb{Z}\times(\mathbb{Z}\backslash\{0\})$ we denote with $r(a,b)$ the remainder of the division between $a$ and $b$. Consider the ...
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487 views

Are ideals in rings and lattices related?

There are (at least) two notions of ideals: An ideal in a ring is a set closed under addition and multiplication by arbitrary element. An ideal in a lattice is a set closed under taking smaller ...
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805 views

Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain.

Let $(X, \leq)$ be an infinite poset. Prove that there is an infinite $X' \subset X$ for which holds one of the following: 1) Induced order on $X'$ is linear. 2) $(X', ...
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3answers
265 views

Is there a name encompassing both limit inferior and limit superior

Is there a mathematical term which would include both liminf and limsup? (In a similar way we talk about extrema to describe both maxima and minima?) The only thing I was able to find was that some ...
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What elements of the pentagon lattice $N_5$ do not satisfy the distributive law?

I heard that the "pentagon lattice" $N_5 = (\{0,a,b,c,1\},\le)$ is not distributive, where $\le$ is the reflexive transitive closure of $\{(0,a),(a,b),(b,1),(0,c),(c,1)\}$. What elements of $N_5$ do ...
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205 views

A certain family of complex functions

Are there an uncountable linear order $(P, \leq)$ and a family of continuous complex-valued functions $\{f_p\colon p\in P\}$ defined on the interval $[0,1]$ such that for any $p<q$, $p,q\in ...
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1answer
144 views

Connected subsets of lines

For me every connected set has at least two elements. Connected subsets of the real line have non-empty interiors. I am curious if the same is true for connected subsets of any connected linearly ...
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Two order related questions for subgroups of infinite groups

For finite groups what I am asking is really trivial. In finite groups if we pick a non trivial subgroup $H$ of $G$ we can always find (there could be multiple choices) a subgroup that is an immediate ...
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1answer
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How to show the ascending chain condition

Let $(A,\le)$ be a poset. Suppose that for any $a < b \in A$ and for any chain $Q$ of $A$ whose maximum and minimum are $a$ and $b$ respectively, $Q$ is finite. Let $C$ be the set of all chains of ...
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61 views

Minimum size of a subset to know a complete total order

Lets say we have a set $A$. Suppose that $A$ is ordered by $<$, $A$ is completely ordered. $<$ can be defined as $<:=\{(a,b) \in A\times A : a<b \}$ Given that $<$ is transitive, it ...
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1answer
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What is a binary relation like whose reflexive transitive closure is a partial order?

Let $R$ be a reflexive binary relation and $R^*$ be its reflexive transitive closure. The question is what is the equivalent condition in terms of $R$ to $R^*$ being a partial order. Intuitively, a ...
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129 views

“Down-Closed”, “Down Ideal”, Something Else?

Let $X$ be an a set and let $\mathcal{C}$ be a collection of subsets of $X$ satisfying the following property: If $A$ and $A^\prime$ are subsets of $X$ with $A \in \mathcal{C}$ and $A^\prime ...
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1answer
181 views

How different are inductively ordered set from posets that satisfy the ascending chain condition?

I'd like to show that posets that satisfy the ascending chain condition (ACC) have maximal elements. I noticed this statement looks much like Zorn's lemma which holds for inductively ordered sets. ...
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647 views

Totally ordering the power set of a well ordered set.

Let's say I take a set $S$, where $S$ can be well ordered. From what I understand, one can use that well ordering to totally order $\mathscr{P}(S)$. How does a body actually use the well ordering of ...
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309 views

The “canonical” representative of an order type

An ordinal (in von Neumann sense) can be thought as the "canonical" representative of the order type (i.e. of the class of all order-isomorphic sets) of a well-ordered set. This is very convenient, ...
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1answer
128 views

Questions about power sets and their ordering

Okay, so I'm stuck on a question and I'm not sure how to solve it, so here it is: In the following questions, $B_n = \mathcal{P}(\{1, ... , n\})$ is ordered by containment, the set $\{0,1\}$ is ...
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Is it possible to construct an ordered field which is also algebraically closed?

It is well known that while the real numbers are totally ordered, they are not algebraically closed, and while the complex numbers are algebraically closed, they are not totally ordered. Is it ...
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How to understand $\limsup$/$\liminf$ of a subset of a complete lattice with a topology

From Wikipedia: Let $Y$ be a partially ordered set which is also a topological space and a complete lattice so that the suprema and infima always exist. For a set $X ⊆ Y$, define $$ ...
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Is every suborder of $\mathbb{R}$ homeomorphic to some subspace of $\mathbb{R}$?

Let $X \subset \mathbb{R}$ and give $X$ its order topology. (When) is it true that $X$ is homeomorphic to some subspace of $Y \subset \mathbb{R}$? Example: Let $X = [0,1) \cup \{2\} \cup (3,4]$, ...
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1answer
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A question about lattices and their being partially-ordered sets, rather than total-ordered sets

Am I correct in saying that, given a and b are elements of S, the join(a, b) and meet(a, b) won't be either a or b, if there is no order between a and b?
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Are all subspaces of the rationals order topologies (with respect to some linear order)? All closed subspaces?

This is a follow-up to this question. Let $Y \subset \mathbb{Q}$, and give $Y$ the subspace topology. Is there necessarily a linear ordering $<$ of $Y$ (possibly different from the order inherited ...
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If $X$ is an order topology and $Y \subset X$ is closed, do the subspace topology and order topology on $Y$ coincide?

Let $(X,<)$ be a linearly ordered set and give $X$ its order topology. Suppose that $Y \subset X$. There are 2 sensible ways to topologise $Y$: View $Y$ as a subspace of $X$ and give it the ...
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256 views

Create an order relation over the field of p-adic numbers

I've come to know that you can't define an order relation over the field of p-adic numbers that is compatible with the addition and multiplication according to the ordered field axioms. I was ...
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Is $\mathbb R$ terminal among Archimedean fields?

I was wondering why metrics and norms are always defined to be real, rather than generalized to some other fields (or whatever). The best guess I have so far is: Because every Archimedean ordered ...
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Identifying maximal, greatest elements on a Hasse/lattice diagram?

So, while I understand the difference between maximal elements and greatest elements, I'm having trouble understanding how to identify them on a lattice diagram (just as an example, the lattice ...