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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Stochastic Orders

Does the partial order below (R) has a name? Has anyone seem something somewhat similar or related before? The book Stochastic Orders (my bible for this stuff) has no reference on it. Any references ...
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How is greater than defined for real numbers? [closed]

Is there a formal definition of greater than? I need it to describe how much better I am than my friends at math. EDIT: I would like to clarify this is partially a joke, and partially a serious ...
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Order-preserving function

I have an order on $\mathbb R ^ \mathbb R: f \le g $ iff $\forall x \in \mathbb R: f(x) \le g(x)$. Now I have a function $\mathbb R ^ \mathbb R \to \mathbb R ^ \mathbb R: F (f)(x)= f(x^2)+1$. I have ...
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Nomenclature for posets s.t. for all $x<y$, there exists $z$ with $x<z<y$.

I can't remember the nomenclature (if it exists) for a poset with the property in the title, that is, a poset $(P,\leq)$ with the following property: If $x<y$ in $P$, then there exists $z\in P$ ...
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The size of largest antichain to the total number of incomparable elements

Given a poset $>$ over a set $A$, every two elements $x,y\in A$ stands in exactly one of three cases: either $x>y$ or $y>x$ or $x\bowtie y$. The last case says $x$ and $y$ are incomparable. ...
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cardinality of maximum antichains in power set posets

Let $\mathcal{P}(S)$ be the power set of a non empty set $S$. Consider the poset $\succ$ for the inclusion relation over the elements of $\mathcal{P}(S)$ (which is equivalently represented by a single ...
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Term of partially ordered set with “levels”

Suppose that we have a partially ordered set $(X,\leq)$ such that the following condition holds: There exists a disjoint partition $X = \bigcup_{ i \in \mathbb N_0 } X_i$ such that for $i < j$ we ...
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Monotone actions of the infinite cyclic group

For reasons which are too long to explain, I grew interest in the theory of monotone actions of partially ordered groups, by which I mean monotone functions $G\times P\to P$ satisfying the axioms of ...
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How ae these three order types on $Z_+ \times Z_+$ different?

Let $Z_+$ denote the set of positive integers. Consider the following relations on $Z_+ \times Z_+$: The dictionary order; that is, $(x_0,y_0) < (x_1,y_1)$ if either $x_0 < x_1$, or $x_0 = ...
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Example of a poset with elements not included in a maximal element?

I am reading Gratzer's General Lattice Theory and one of the exercises is to give an example of a poset with maximal elements in which not every element is included in a maximal element. I am ...
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Order Type Stratification

Is there some sort of interesting way of organizing certain order types that aren't ordinals? When I say certain order types, some examples include but are not limited to: order types of dense ...
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Uncountable Dense Linear Orders

Is there an example of two uncountable equipollent dense linear orders without endpoints that don't satisfy the same first order properties? Or is it true that two uncountable equipollent dense linear ...
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Order preserving functions that do not preserve binary operations

According to Tarski's Fixpoint Theorem for lattices, if I have a complete lattice, $L$, and an order-preserving function, $f:L \to L$, then the set of all fixpoints of $L$ is also a complete lattice. ...
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Well ordering+finite number of steps between elements property name?

So, I just did a counterexample in topology that relied upon a well ordering in which you there were a pair of elements that you could not connect between in a finite number of steps. (The standard ...
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Prove if 0(a) is odd, show a is a square.

Let $G$ be a group and a is an element of $G$. If $o(a)$ is odd, show $a$ is a square. I started by supposing that $o(a) = 2n+1$ which implies $a^{2n+1}=1$. I am not sure if I am on the right track ...
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An interesting puzzle for some, confusing for me

Suppose that $a$ is of odd order $k$ and $bab=a$. I need to show that $b$, must be of order $2$. We can prove this anyway we want to, but our hint is to expand $(bab)^k$ and re-associate and then ...
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The “Largeness” of Subsets of the Natural Numbers

Recently, I was thinking about the Erdős conjecture on arithmetic progressions, which says that, if, for a set $A\subseteq\mathbb{N}$, the sum $\sum_{a\in A}\frac{1}a$ diverges (i.e. "A is large"), ...
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Non-continuous Poset without Interpolation property.

There is a theorem that said if $X$ is a continuous poset, then if $x \ll y$, there exists $z\in X$ such that $x\ll z\ll y$, where $\ll$ is the way below relation. I am not sure why the continuity ...
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Total ordering on the free group

The free groups can be totally (bi-)ordered. This paper shows how to do it (page 4). In short, you embed the group in multiplicative structure of the ring of power series in non-commuting variables, ...
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For real number, Is it true that $sup\{x_i + y_i: i\in I\}=sup\{x_i:i\in I\}+sup\{x_i:i\in I\}$?

If $I$ is an indexing set for real numbers so $x_i \text{and} y_i$ are reals. Is it true that $sup\{x_i + y_i: i\in I\}=sup\{x_i:i\in I\}+sup\{x_i:i\in I\}$? I thought I need this statement to prove ...
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Can someone give me some clue on how to show that rationals are well ordered? Thank you in advance.

I wanted to show that the set of rationals are well ordered. A small hint would be really appreciated.
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Order between probability measures: sets full below

Consider a product space $X = \{0,1\}^\mathbb{Z}$ and the space of probability measures on $X$, $\mathcal{M}(X)$. We say that for any two $a, b \in X$, $$a \prec b \iff a_x \leq b_x \, \, \, \, \, ...
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Show that a relation is a partial order with lubs and glbs of all pairs

I came across this question while doing my discrete mathematics and couldn't understand it completely. The question is as follows: Let $(N, ≤) $ be the set of natural numbers with the relation $ m ≤ ...
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Any total order on $X$ is maximal element

Let $X$ be a non-empty set and $R$ be the set of partial orders on $X$. (1) Show that $R$ is partially ordered by inclusion $\subset$. (2) Show that any total order on $X$ is a maximal element in ...
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On the definition of Scott Continuity

I somewhere encountered the concept of "Scott Continuity" as follows. Let $P,Q$ be partially ordered sets; a function $f:P\to Q$ is Scott continuous if it preserves directed suprema, i.e. for all ...
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In the context of posets, is the notion of “compatible” elements a standard notion?

Let $P$ be a poset and let $x$ and $y$ be elements of $P$. I have occasionally seen books define a notion of "compatible" elements in a poset, so that $x$ and $y$ are compatible iff $\exists a \in P$ ...
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Extend an acyclic relation to an ordering

I am pre-studying a course (Discrete Mathematics) that I will be taking come fall quarter this October. We are using the textbook Invitation to Discrete Mathematics and I am having trouble starting to ...
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What is the supremum of {36,72} in this Hasse-diagram?

In this Hasse-diagram is sup({36,72})={72} or is it non-existent? I think it might be {72} because {72} seems to be the upper bound of {36,72}. Then again, {72} seems to be the only upper bound, so ...
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Polynomial time algorithm for determining if there exists an ordering of subsets

Given n subsets of cardinality k of a set $S=\{1,2,...,m\}$. Is there a polynomial time algorithm to determine if there exists an ordering of subsets $s_1,...,s_n$ such that ...
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Automorphism of $(\mathbb{R}, <)$ that takes a finite set of reals to a set of naturals

I was working on another of Enderton's logic book exercises (specifically, exercise 20 (b) from section 2.2, p. 102), and, as part of the exercise, he asks us to show that, for any finite set of real ...
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Alternative definition for order-embedding

I realize that the traditional definition for an order-embedding f is that $a_1 \sqsubseteq a_2 \iff f(a_1) \sqsubseteq f(a_2)$ However, is it also fair to say that if $(A, \sqsubseteq)$ is a ...
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If a linearly ordered set $L$ has the property that every order-preserving injection $L \rightarrow L$ is expansive, is $L$ necessarily well-ordered?

Given a poset $P$, call a function $f : P \rightarrow P$ expansive iff $f(x) \geq x,$ for all $x \in P$. Now suppose a linearly ordered set $L$ has the property that every order-preserving injection ...
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Is $\{1,2\}\times [0,1)$ a linear continuum in the dictionary order?

In Munkres' Topology, it's stated that if $X$ is well-ordered set then $X \times [0,1)$ is a linear continuum in the dictionary order. Let us suppose that, $X=\{1,2\}$ and that $\mathord{<} ...
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For any $n \in \mathbb{N}$ there exists a minimal element in the set $ \{ x\in \mathbb{N}\mid n< x \}$

Problem from V. Zorich Analysis textbook: For any $n \in \mathbb{N}$ there exists a minimal element in the set $ \{ x\in \mathbb{N}\mid n< x \}$ namely $\min\{x \in \mathbb{N}|n<x\}=n+1$* ...
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If a poset is Dedekind-complete, is its opposite necessarily Dedekind-complete?

Call a poset $X$ Dedekind-complete iff for all non-empty sets $A \subseteq X$, the following are equivalent. $A$ has an upper bound $A$ has a least upper bound. Supposing $X = (X, \leq)$ is ...
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Constructing order embeddings between Boolean algebras from embeddings from their finite subalgebras

Suppose that $A$ and $B$ are two complete atomic Boolean algebras and $R$ is a relation between $A$ and $B$ with the following property: If $Rab$ and $A^\prime$ is a finite Boolean subalgebra of ...
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Prove that the binary relation “is a subset of” is a…

Prove that the binary relation "is a subset of" is a partial order (POSET)? Should I try to prove this in reference to the power set of a general set? When is this relation a total order?
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An accessible example of a preorder that is neither symmetric nor antisymmetric

For a project I am working on, I need an example of a preorder (reflexive and transitive relation) that is neither symmetric (like an equivalence relation) nor antisymmetric (like less than or equal ...
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Is it Preference relation?

I need to check if the relation $ \succeq ( \space \succeq \space \subset X × X, \space X=VB[0,1] ) $ define as below $$ f \succeq g \Longleftrightarrow Var(f+g) \geq Varf $$ is preference ...
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Well-ordered family of open sets.

Let $X$ be a second countable space. I $\textbf{A}$ is a family of open set well-ordered by inclusion prove that this family is numerable. I have the following idea: Let $\textbf{B}$ be a ...
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A question about the cardinality of a set of functions with finite support where the domain of each function has cardinality aleph-null.

Suppose that $M$ is a well ordered set with at least two elements and that $L$ is a well ordered set with the same cardinality as the natural numbers. Let $[L, s.f, M]$ be the set of all functions ...
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Dilworths Theorem proof doubt

This is the proof I am talking about http://www.math.cmu.edu/~af1p/Teaching/Combinatorics/F03/Class14.pdf When you take a maximal chain C in P and then obtain antichains in P\C, if the size of the ...
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Correctness of proof that every positive rational with square $>2$ is an upper bound for those with square $<2$

I would like to know whether my proof makes sense or not, and if not where should it be corrected. Let $E=\{x \text{ is rational }: x>0 \text{ and } x^2<2\}.$ Claim: Every member of $F=\{x ...
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Every subgroup of a cyclic group is characteristic (using lattice theory).

I want to show for $n\in\Bbb N$, which is not square-free, that every subgroup of $Z_n=\langle x\;|\;x^n\rangle$ is characteristic. But I want to show it in a convoluted way. Every automorphism ...
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Are $1+ω$ and $ω+1$ isomorphic?

In set theory, $1+ω$ is defined as the ordinal number of ordinal sum of sets $\{a\}$ and ℕ. Also $ω+1$ is defined as the ordinal number of ordinal sum of sets $\Bbb N$ and $\{a\}$. You know what the ...
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Let $L$ and $L'$ be lattices. Prove that $L \times L'$ is also a lattice.

Can someone please verify my proof or offer suggestions for improvement? Some preliminaries: Let $A$ and $B$ be two posets. $A \times B$ is the poset on the cartesian product of $A$ and $B$ such ...
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Prove that $P$ is a lattice (details inside)

Can someone please verify my proof or offer suggestions for improvement? There may be answers to the same questions elsewhere, but I need help with my proof in particular. Show that if $P$ is a ...
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Let $P$ be a finite poset. Show that the number of order ideals equals the number of antichains.

Can someone please verify my proof or offer suggestions for improvement? I am aware that there are similar questions posted elsewhere, but I need help with my proof in particular. Some preliminaries: ...
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Let $P, Q, R$ be finite posets. Prove that $P^{Q+R} \cong P^Q \times P^R$.

Can someone please verify my proof and offer suggestions for improvement? I feel that my proof might have been a little hand-waving in showing that $\varphi$ is a bijection, and I feel that it is not ...
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Is the function $x \mapsto \max\{f(x),g(x)\}$ always an automorphism if $f$ and $g$ are?

Consider a totally ordered set $X$ and a pair of automorphisms (i.e., order-preserving bijections) $f,g$ of $X.$ Then the function $$h:x \in X \mapsto \max\{f(x),g(x)\} \in X$$ is an order-preserving ...