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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correct English pronunciation of the word poset

What is the correct English pronunciation of the word poset (Partially Ordered SET)? paazit or pow-set?
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Any countable set $A$ with a dense total ordering, with neither a maximum nor a minimum, is isomorphic to the rationals.

I think I can do this first: index all elements of $A$ by the natural numbers, make a map $f$, first send $a_0$ to any rational number, call it $q_0$. Then inductively, depending on the ordering of ...
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$\forall{a\in A,\, b\in B} \mid a\le b$ Prove that: $\sup A \le \inf B$

Let A and B be two non-empty bounded subsets of $\mathbb{R}$. $\forall{a\in A,\, b\in B} \mid a\le b$ Prove that: $\sup A \le \inf B$ My solution goes as follows: Suppose $\sup A \gt \inf B$: ...
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Ways Of Ordering A Set

Perhaps this is a strange question. But it made me think and I have no idea how to answer it. Or if it actually makes sense. I've come across "Well-Ordered Sets" a few times now, and it's well known ...
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220 views

The relation between order isomorphism and homeomorphism

Let $X$ be a set and let $<_1,<_2$ be order relations on $X$. Let $T_1,T_2$ be the topologies induced on $X$ respectively. If $(X,T_1)$ is homeomorphic to $(X,T_2)$, does that imply that ...
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Different orderings for highest weights of a representation

Recall that given a representation $\pi$ of $\mathfrak{sl}_n$, a weight $\mu$ is said to be of highest weight if its corresponding weight vector is annihilated by all the positive root spaces (1). ...
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167 views

Discrete Math “Well Order”

the definition of a well order is that if $R$ is a linear (order) and every non-empty subset of $A$ has a least element. I understand that $(\mathbb N,\le)$ is a well-order but how come $(I,\le)$ ...
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Is the set $S$ of functions $S = \{ f : \mathbb R \to \mathbb R \}$ from the reals to the reals a totally ordered set?

Is the set $S = \{ f : \mathbb R \to \mathbb R \}$ of functions $f$ from the reals to the reals a totally ordered set ? I think the question is clear. Note that there is a bijection to $g(x)$ gives ...
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Orders on the Cartesian product of partially ordered sets

The wikipedia article about partially ordered set describes three of the possible partial orders on the Cartesian product of two partially ordered sets, and mentions that similar constructions can be ...
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Determining minimal and maximal elements in products of partial orders

$\newcommand{\angles}[1]{\langle { #1 } \rangle}$ I did the following exercise, can you tell me if I got it right? Thanks: EXERCISE 31(G): Consider $\angles{ [0,1) , \leq } \otimes^s \angles{ ...
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Showing that the equivalence class of partial orders order isomorphic to $(X, \leq)$ is not a set

I'm trying to do the following (original image): EXERCISE 25(R):(a) Argue on the grounds of the architect's view of set theory (which was outlined in the Introduction) that $\mathbf{PO}$ cannot be ...
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Proof of every binary relation with a rank function is well-founded

I proved the following claim (source image), can you tell me if my proof is correct? Thanks: Claim 4: Let $R$ be a binary relation on a set $X$, and suppose $\langle Y, \prec \rangle$ is a strict ...
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How to define a well-order on $\mathbb R$?

I would like to define a well-order on $\mathbb R$. My first thought was, of course, to use $\leq$. Unfortunately, the result isn't well-founded, since $(-\infty,0)$ is an example of a subset that ...
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76 views

Disjointifying sets if indexed by set that doesn't admit a well-order?

I have a question about the following: I think this should say "... if $I$ finite or if there exists a well-order ..." because if $I$ is a set like this also lets one disjointify $A_i$ with $i ...
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Proof of “well-founded iff there is no sequence $x_n$ s.t. $(x_{n+1}, x_n) \in R$”

I tried to do the following exercise from a book I'm reading, can you tell me if my solution is correct? Thanks! Theorem 2: Let $R$ be a binary relation on a set $X$. Then $R$ is strictly ...
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A question about infinite utility streams

At the end of Diamond's Evaluation of Infinite Utility Streams he proves a theorem (which he doesn't give a name to, but it's at the very end of the article). There is a step in which he jumps from ...
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Relationships between zero morphisms and least morphisms

Zero morphism $0_{XY}$ is defined by the formulas $a\circ 0_{XY}=b\circ 0_{XY}$ and $0_{XY}\circ c= 0_{XY}\circ d$ for every morphisms $a$, $b$, $c$, $d$ of suitable sources and destinations. I ...
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Relation between lattice theory and semilattice theory

When studying inverse-semigroups, idempotent elements play an important role, and semilattices occur naturally as sets of idempotent elements that commute with each other. I have the impression that ...
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155 views

Shortest proof of Kuratowski's lemma

I want a short proof of Kuratowski's lemma (about maximal chains). Does proving Kuratowski's lemma need first prove Zorn lemma? I am writing a book in which I claim that the reader needs to know ...
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338 views

Is this a partial order and draw the Hasse diagram.

$A=\{2,3,4,6,8,12\},\ x,y \in A, x\le y \leftrightarrow x^2 \mid y$ Is this a partial order and draw the Hasse diagram. I know to be a partial order it needs to be reflexive, anti symmetric ...
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Isomorphism of preferences

Suppose we have a denumerable set $S$ with a total order $\preceq$. Is there necessarily an isomorphism $\phi:S\to\mathbb{N}$ such that $x\preceq y \Leftrightarrow \phi(x)\leq\phi(y)$? Specifically ...
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361 views

Cross Product of Partial Orders

im going to have a similar questions on my test tomorrow. I am really stuck on this problem. I don't know how to start. Any sort of help will be appreciated. Thank you Suppose that (L1;≤_1) and ...
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$\sup(S)$ does not belong to $S$?

Problem: can anyone come up with an ordering of $\mathbb{N}$ different than the standard one we know, where we can find a subset $S\subset \mathbb{N}$ in a way such that $\sup(S)$ exists in ...
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How to find a total order with constrained comparisons

There are 25 horses with different speeds. My goal is to rank all of them, by using only runs with 5 horses, and taking partial rankings. How many runs do I need, at minumum, to complete my task? As ...
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What does a well ordering of $\mathbb{R}$ look like? [duplicate]

Possible Duplicate: Is there a known well ordering of the reals? I am having a hard time wrapping my head around what a well-ordering of $\mathbb{R}$ looks like. I have seen the ...
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Is the semigroup generated by wellordered positive set wellordered?

Let $(A,\leq)$ be a totally ordered abelian group, and $\Gamma\subseteq A$ be a set of nonnegative elements, such that it is wellordered by $\leq$. Is it true then that the semigroup $S$ generated by ...
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133 views

Equality on Partial and Total Orders

If I have a partial order, is the following conclusion valid? $$a \geq c$$ $$b \geq c$$ Then, $$a = b $$ Does the result change if the partial order were to become a total order? Any and all ...
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How to define subtraction in lattice and how to define complementation of an element in a lattice?

The story is that in a lattice, there are some elements incomparable, and there exits some elements have overlapping elements. for example, suppose the partial order is defined on inclusion, then ...
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How to prove that the infimum of the sum of two sequences is at most the sum of their infima?

I'm trying to show: $\inf(x_k+y_k) \leq \inf(x_k)+\inf(y_k)$. So I did: Let $r>0$, there is a $k_1$ such that $x_{k_1}\leq \inf(x_k)+r/2$ and there is a $k_2$ such that $y_{k_2}\leq ...
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Poset Infimum and Supremum

I was asked to show that if every subset of a poset has an infimum then every such subset has a supremum. I did my proof and now I realize that what I was calling "infimum" was actually "a smallest ...
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Sorting $i$ amongst integers

In the game Wits & Wagers, players each answer a numerical question and the answers are then sorted in ascending order for scoring, details of which being irrelevant for this question. In this ...
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Conditional merge of a sequence of DAGs / partial orders

I have a sequence of directed acyclic graphs, $G_i$. The union of their edge sets may not be a DAG. I want to find an edge set that is maximal in some sense but still acyclic. Alternatively I have a ...
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Generalization of ordinals to well-founded sets?

In set theory, the ordinal numbers are objects that represent the order types of well-ordered sets. Is there a similar class of objects that represent the order types of well-founded sets? If so, ...
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Number of linear extensions of a finite partially ordered set

Let $N$ be a finite set and $\le$ be a partial order. Can you express the number of linear extensions $L(\le,N)$ of this partial order in terms of $b(k) := \#\{l \in N\ :\ k \le l\}$? My intuition ...
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Continuation of strictly monotone functions on $\mathbb{R}$

While studying the properties of ordinal utility functions, I came across the following question. Given a strictly increasing function $f : D \rightarrow \mathbb{R}$, where $D$ is an arbitrary ...
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Atoms necessary for the existence of a generic filter?

I'm reading Some applications of the method of forcing by Todorchevich and Farah and one of the exercises seems to be wrong to me. Definitions We fix some partially ordered set $(P,\preceq)$. A ...
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Problem with “tree” definitions

In my studies of choice principles, I've encountered the concept of a tree several times. Frustratingly, no two of the sources I've been working with define it in quite the same way! Notation: Given ...
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Infinite linear (non-well) orderings

I can rather easily imagine the infinite linear (non-well) orderings $(\mathbb{Z},\leq)$, $(\mathbb{Q},\leq)$, $(\mathbb{R},\leq)$, each one of its own order type. Are there "essentially" other ...
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How to derive distributivity from Boolean algebra laws

Let $(L,\le,\bot,\top)$ be a bounded lattice and $\neg: L \rightarrow L$ be a map that satisfies the following laws: $a \wedge b = \bot \Leftrightarrow a \le \neg b$ $\neg\neg a =a$ I'd like to ...
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If $\sim$ is an equivalence relation on $X$, and there is a strict total order on $X/\sim$, what kind of ordering does $X$ have?

I would like to know if there's a special name for this kind of ordering. When I say there is a strict total order on $X/\sim$, what I mean is that two distinct elements in the same equivalence ...
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What is the difference between the supremum (least upper bound) and the minimal upper bound

I understand the difference between the supremum and the greatest element and between the supremum and the maximal elements. But I'm not sure about the difference between the supremum (least upper ...
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Why is 'Antisymmetry' named so?

So when we talk about order relations for the familiar number systems, we are always introduced to the antisymmetry property which is $x \le y, x \ge y \implies x=y$. When I think of the word ...
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Totally ordered set with greater cardinality than the continuum

Does there exist a totally ordered set $S$ with cardinality greater than that of the real numbers? Sequences are continuous functions with domain $\mathbb{N}$ and paths are continuous functions with ...
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Congruence for embeddings of a poset into semilattices

Let $\mathfrak{A}$ be a poset, $\mathfrak{B}$ and $\mathfrak{C}$ be meet-semilattices with least elements. Let $f:\mathfrak{A}\rightarrow\mathfrak{B}$ and $g:\mathfrak{A}\rightarrow\mathfrak{C}$ are ...
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Every poset is embedded into a meet-semilattice

I "discovered" a few minutes ago that every poset can be embedded into a meet-semilattice. Let $\mathfrak{A}$ be a poset. Then it is embedded into the meet-semilattice generated by sets $\{ x \in ...
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preorders induced by continuous functions to the reals

Any function to a (total) order induces a (total) preorder on its domain. What can be said about the total preorders induced by a continuous function from a "nice" topological space (for instance, a ...
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Archimedean ordered fields and valuations (exercise)

Definitions: Let $(K,\leq)$ be a totally ordered field and $(G,\leq)$ a totally ordered abelian group (written additively). If we denote $\mathbb{Z}a\!=\!\{na;\, n\!\in\!\mathbb{Z}\}$, then $G$ is ...
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Draw Hasse diagram as two elements have same image

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$, $P=\{y \in \mathbb N : y \text { is a prime number}\}$, consider the map $f$ defined as follows: $$\begin{aligned} f:x\in S \rightarrow f(x) \in \wp (P) ...
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What is a valuation associated to an ordering on a field?

If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined? I was searching through Prestel & Delzell's Positive ...
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Names of certain morphisms in Pos

Pos is the category of small posets and monotone maps. I call a morphism $f:\mathfrak{A}\rightarrow\mathfrak{B}$ of Pos monovalued iff it maps every atom of $\mathfrak{A}$ either into an atom of ...