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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Is it still possible for their to exist a well-ordering on the reals such that there is always a least next element?

Let $a \leq b$ be a well-ordering on $\Bbb{R}$. And suppose for any $a_n \in \Bbb{R}$ there exists another element $a_{n+1} \in \Bbb{R}$ such that under the given ordering $(a_n, a_{n+1})$ is empty. ...
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Self-duality in a lattice

Is there any finite self-dual lattice $(X,\le)$ such that there is not any self-duality $f:X\to X$ such that $f\circ f = 1_X$? Let $f,g:X\to X$ be a self-dualities. Then $f^{-1}\circ g$ is an ...
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Partitioning a totally ordered set into three subsets according to the order

Consider a set $S$, and a total order $R$ over that set. Part (a) Given some element $e \in S$, explain why it is possible to partition $S$ into the following three sets: $$S_1 = \{ x \in S ...
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Proving that a poset can be expressed as a union of subchains.

Let $L$ be a finite partially ordered set where the maximum size of an antichain is $n$. I need to prove that $L$ can be expressed as a union of $n$ subchains. Here is my attempt: $L$ must have at ...
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Proving that a convex subset of a chain is the intersection of a lower segment and an upper segment of a chain.

The book I am using is very naive so AC is assumed. Call a subset $S$ of a chain $L$ a lower segment if: $x \in S$ and $a < x$ implies $a \in S$. An upper segment is defined equivalently. Call a ...
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Find the Fixed points (Knaster-Tarski Theorem)

Let $L=\mathcal P(\mathbb N)$ be a complete lattice of subsets of $\mathbb N$. a) Justify that the function $F(X)=\mathbb N \setminus X$ does not have a Fixed Point. I don't know how to solve this. ...
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1answer
14 views

Equivalence between different definitions of transitive closure

Let $W$ be an arbitrary, non-empty set and let $R$ be an arbitrary binary relation on $W$. Define the transitive closure of $R$ as $R^+ = \bigcap \{ R' \; | \; R' \text{ is a binary transitive ...
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186 views

A reflective subcategory of the category of inverse semigroups.

The Question. I'm reading Lawson's "Inverse Semigroups: The Theory of Partial Symmetries" and I've hit something I don't understand. It's claimed on page 34 of my copy that The category of ...
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Maximal Element vs Greatest Element [closed]

Disclaimer: This thread is written in Q&A style. The answer is provided below. Let $A$ be a poset. If $a$ is a greatest element then $a$ is a unique maximal element. Is the converse true as ...
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Well-ordering principle and negative integers

The Wikipedia article on the Well Ordering Principle defines it [1] as: "The well-ordering principle states that every non-empty set of positive integers contains a least element." And it defines ...
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Why we use ANY in the definition of a maximal element?

I am confused about the following definition: "a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S." I do not understand ...
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In a lattice, does $x \vee y \leq x\vee z$ and $x \wedge y \leq x\wedge z$ imply $y\leq z$?

Let $L$ be a lattice and $x,y,z\in L$. If $y \leq z$, then clearly $x\vee y \leq x\vee z$ and $x\wedge y \leq x\wedge z$. Now I wonder about the reverse direction. In general, $x\vee y \leq x\vee z$ ...
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1answer
33 views

About definition of lexicographical order

Def. let be $\preceq_A$ a total order, $(a_1,a_2,...,a_n),(b_1,b_2,...,b_n) \in A^n$, $(a_1,a_2,...,a_n) \leq^d (b_1,b_2,...,b_n)$ if one and only one of the following holds is true: $$a_1 \prec_A ...
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2answers
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Why do 42 and 30 have the same structure in the diagramm of Hasse?

I have an exam coming up tomorrow and there's just one more question more to prepare. I would be so gratefull if anyone could help. It´s about the relations of dividers. By the help of the diagram of ...
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Is this connected relation?

My task is to check if this is preference relation (connected and transitivited) $$ f \succeq g \Leftrightarrow \forall x\in [0,1] f'(x) \leq g'(x) $$ My solution is: that this relation is not ...
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A partial order with more properties than would be expected

Consider the relation: $$\langle x_1,x_2\rangle\prec\langle y_1,y_2\rangle\iff x_1,x_2,y_1,y_2\in\Bbb N\wedge x_1y_2<x_2y_1.$$ This is usually used for defining the (positive) fractions $\Bbb ...
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How to define an isomorphism between $^\omega\omega$ and $\omega^\omega$?

Let $^\omega\omega$ be the set of all functions $x: \omega \to \omega$. Define $A = \{x \in ^\omega\omega \; | \; x \text{ has finite support}\}$, where by "finite support" I mean that the set $\{x(n) ...
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Order Preserving Isomorphism

If abelian group G has an archimedean order then there is an order preserving isomorphism $\phi$ of G onto a subgroup of $\mathbb{R}$. Here we can say that G is archimedean totally ordered abelian ...
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Bijection from $\mathbb{R}^n$ to $\mathbb{R}$ that preserves lexicographic order?

Is there a bijective mapping $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ that preserves lexicographic order? That's to say, we'd need to have $f(x_1, \dots, x_n) \leq f(y_1, \dots, y_n)$ iff ...
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Existence of certain uncountable closed sets in the order topology

This is a proof-verification request. Let $\Omega$ be the set of countable ordinals, $\omega_1$ the first uncountable ordinal, and $\Omega^*=\Omega\cup\{\omega_1\}$. Remarkable properties of these ...
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Proof on Dyadic Trees [Smullyan: First-Order Logic, chapter 1, section 0]

I'm having difficult with a proof from Smullyan's First-Order Logic, Chapter 1 Section 0 (Reprint, Dover 1968, p. 4): Prove: In a dyadic tree, define x to be to the left of y if there is a ...
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1answer
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Does a discrete semilattice of finite width have finite number of points between any pair of elements.

Let $(S,<)$ be a countably infinite semilattice such that: 1) $S$ is no-where dense - i.e. there does not exist a subset $T$ such that for all $a,b\in T$ with $a<b$, there exists $c\in T$ with ...
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Non-monotonic functions on ordered sets

I'm trying to prove that if $~~(A,<_A)~~$ and $~~(B,<_B)~~$ are linearly ordered sets and $~~f: A \rightarrow B~~$ is non-monotonic function than there exist points $~~a,b,c\in A~~$ such that ...
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43 views

comparing two sets in set theory

I have two sets A and B and this condition holds: $\forall x \in A , y \in B: x \leq y$ Is there any standard term to describe the relation of A and B? something like $A \leq B$? Thanks for your ...
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Undistinguishable elements in posets

Given a finite partially ordered set $P = (V, <)$, I say that $x$ and $y$ in $V$ are indistinguishable in $P$ if for all $z \in V \backslash \{x, y\}$, I have $z < x$ iff $z < y$, and $x < ...
3
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Definition in Kunen

In Kunen's second edition of set theory he gives the following definition Let $(\mathbb{Q},\leq_\mathbb{Q},\mathbb{1}_\mathbb{Q})$, and $(\mathbb{P},\leq_\mathbb{P},\mathbb{1}_\mathbb{P})$ be forcing ...
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Is a complete flag well-ordered?

Given a vector space $V$, a collection $\mathcal{F}$ of subspaces of $V$ is called a flag of $V$ if $\{0\}\in\mathcal{F}$ $V\in\mathcal{F}$ $\mathcal{F}$ is a chain Furthermore, a flag ...
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How to make an order isomorphism

Two linear orders $A$ and $B$ have starting points $a_0$ and $b_0$, and have cofinalities $\omega_1$. Let $(a_\alpha )_{\alpha<\omega_1}$ and $(b_\alpha )_{\alpha<\omega_1}$ be cofinal ...
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1answer
51 views

How to determine distribution

I hope you will be patient with the inarticulate question of a non-mathematician. It's hard to get an answer when you don't even know how to ask the question. But here goes... ...Actually, I have two ...
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3answers
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Name the property $f(x) \ge x$

It's a really one of the simplest properties you could imagine for a function. But I haven't been able to find a name for it. What do you call a function $f$ with the following property: $$f(x) \ge ...
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1answer
58 views

Lower bounded lattice to complete lattice

My problem is to show that any lower-bounded lattice satisfying the maximal condition is a complete lattice. Let's call the lattice $L$. I'm having some trouble with this. I have tried to look at it ...
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Profinite completion of a partial order

In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...
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Suprema always exist $\iff$ Infima always exist

Let $X$ be a poset (or only preordered or even just equipped with a plain relation). Is it true that suprema always exist iff infima always exist: $$\left(\forall A\subseteq X: \sup A\text{ ...
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1answer
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Proof that order-isomorphisms are bijective

I'm reading the Davey and Priestley's Introduction to Lattices and Order. On page 3, it is said that order-isomorphism is necessarily bijective. I thought we could easily prove the claim given the ...
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Set theory, seems like a difficult question (Choice Function)

So, I don't know if this certain type of set has a name in English. If it has, I'll be glad if someone will edit my post. Definition: Let $A$ be a set and $f: P(A)\setminus \emptyset \rightarrow A$ ...
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Defining principal elements of every poset. Is this a new idea?

Fix an arbitrary complete lattice $\mathfrak{A}$ with order $\sqsubseteq$. I call elements $a,b\in\mathfrak{A}$ intersecting and denote $a\not\asymp b$ iff there is a non-least element ...
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1answer
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Cardinality of partial orders on $\mathbb R \times \mathbb R$

Let the set $A$ be defined: $A=\{X \subseteq \mathbb R \times \mathbb R: X$ is a partial order $\land \max X \in \mathbb Z \land min X \in \mathbb Z \}$ What's the cardinality of $A$?
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If an infinite well-ordered set has initial segments of finite cardinality only, is the set isomorphic to $\mathbb N$?

Let $A$ be an infinite well-ordered set. Every initial segment of $A$ is finite. Is $A$ isomorphic to $\mathbb N$? What's the way to think about it? Should I build an explicit isomorphism? What ...
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Understanding first and minimal elements

Give an example of a set with partial order, that has a single minimal element but no first element. The example that was given is $\mathbb Z \cup \{2.5\}$, with relation $<$. I can't see why the ...
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increasing subset of a partial order and characteristic function

Can someone help me understand this? Suppose that $\preceq$ is a partial order on a set $S$ and that $A\subseteq S$. If $\mathbf{1}_A$ is the indicator function then $A$ is increasing if ...
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Is (total-)order a weak version of countability?

One can easily see that countability also imposes or enables an ordering of the set. Now $\mathbb{R}$ is ordered but not countable. Is this ordering a weaker version of countability (at least in ...
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compactness and maximal elements

Let $C$ be a nonempty compact subset of $R^n$, with a certain partial order defined on it. I am trying to prove that $C$ contains a maximal element. My idea is: start with a certain element of $C$. ...
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Compactness and existence of Pareto-efficient cake partitions

I am trying to understand a fundamental statement in the theory of cake-cutting. BACKGROUND: There is a certain "cake" $C$ (a subset of $R^n$). The cake is divided among two agents, 0 and 1. Each ...
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Antisymmetric relation (“strong” vs “weak”)

Defining: "weak antisymmetric relation": $\forall a, b \left< a,b \right> \in R \land \left< b,a \right> \in R \Rightarrow a=b$ "Strong antisymmetric relation": $\forall a, b \left< ...
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Are surjective order-homomorphisms necessarily complete lattice homomorphisms?

Let $X$ and $Y$ denote complete lattice, and suppose $f : X \rightarrow Y$ is a surjective order-homomorphism. Does $f$ necessarily preserve arbitrary suprema, therefore being a complete lattice ...
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Order Type of $\mathbb Z_+ \times \{1,2\}$ and $\{1,2\} \times \mathbb Z_+$

I'm currently working on §10 of "Topology" by James R. Munkres. I've got a problem with task 3: Both $\{1,2\} \times \mathbb Z_+$ and $\mathbb Z_+ \times \{1,2\}$ are well-ordered in the ...
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Divisibilty as relation set on $(\mathbb N \setminus \{0,1\})$

So i have to see if $\prec$ is order relation where two elements $(a,b)$ and $(c,d)$ are in relation $\prec$ if $a|c$ and $2b^{2}+6b\leq2d^{2} + 6 d$. This relation is defined on set $(\mathbb N ...
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Principal order filters on a POSET

I have another problem, but in this one I have no idea how to start. Let be $(X,\leq)$ a POSET with a first element and gifted with the topology $\{ (a,\rightarrow) : a \in X \}$ (principal order ...
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Greatest lower and least upper bounds for a set of pairs

Had some trouble with this question in an exam recently, and wanted to make sure I reasoned correctly. The question was: $X$ is a set of pairs of real numbers $(x,y)$, with absolute values less than ...
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Is this proof legal?

Let $\left(P,\le\right)$ denote a poset. Statement: if every sequence $p_{1}\leq p_{2}\leq\cdots$ in $P$ stabilizes (in the sense that for some $n$ we have $k>n\Rightarrow p_k=p_n$) then every ...