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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Ordered ring and mutiplicative ordinal

If $(E,<)$ is a linear order, let $s(E)$ denote the supremum of the set of ordinals which (order-)embed in $(E,<)$. $s(E)$ is also the set of ordinals which embed in $E$ with a non cofinal ...
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Infinite Trees, Infinite Branches, And Choice

https://en.wikipedia.org/wiki/Tree_(set_theory) Defines a tree T order-theoretically as a poset that has a minimum element R, and for each t in T, the set of lower bounds of T is well-ordered. ...
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Are there results for relations between upward and downward closed partitions of some powerset?

I stumbled upon this, given some set $X$ and its powerset $\mathcal{P}(X)$ and some incomparable set $\mathbb{S}\subseteq\mathcal{P}(X)$, i.e. for any $S,S'\in\mathbb{S}$ we have $S\setminus ...
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45 views

Definability in $\Bbb N$ + $\Bbb Z$

Which elements are definable in $\Bbb N$ + $\Bbb Z$? Where an element, a, is definable if there exists a formula such that $\forall x(\phi(x) \rightarrow x = a) $. I have that all elements of $\Bbb ...
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30 views

How to arrange the following sets?

Given the set $\mathcal{P}(\mathbb{N})$ for the following order: For $A,B \in \mathcal{P}(\mathbb{N}) $ applies $A \leq_{set} B \Longleftrightarrow_{def} A = B$ or$ ~$ min$(A\triangle B)\in B$ We ...
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44 views

A non-separable, connected ordered topological space with no interval order isomorphic to the reals

Is there a non-separable, connected ordered topological space such that for each point $x$ in that space, there is no interval containing $x$ that is order isomorphic to real line?
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41 views

If there are order preserving injections between two countable sets, will there be an order preserving bijection?

During my Topology class the following question was brought up: Question: Suppose that $A$ and $B$ are ordered countable sets. If there are two order preserving injections $f:A \to B$ and $g:B \to ...
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43 views

If there is a simple group of order less than 36, then it must have prime order

I want to show that if there is a simple group of order less than 36, then it must have prime order. Is there a quick way to show this or do I have to go through each order 1 though 36 showing that ...
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13 views

Separative forcings add new sets

Let $\mathbb{M} = (M, \in)$ be a class which models ZFC. A forcing $\mathcal{P} = (P, \leq)$ on $M$ (where $P \in M$ and $\mathbb{M} \vDash P \in M$) is separative if for every $r \in P$ there are ...
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33 views

A map between two totally ordered sets is a homeomorphism if and only if the map is oreder preserving

I'm a little stuck with this problem and I'd appriciate any advice or hint you could give me: Let $(X,\leq)$, $(Y,\preccurlyeq)$ totally oredered sets and $\psi: X \rightarrow Y$ a function. Then: ...
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25 views

intersection on Poset Topology.

Consider $(X,\prec)$ partial ordered set and consider the collection of set $\{U_L(x) \}_{x \in X}$ where, $U_L(x) = \{y\in X: y \prec x \}$ . It is easy to prove this is a base for some topology over ...
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Mobius functions for acyclic categories, question about formula in “Combinatorial algebraic topology” by Kozlov,

Let $C$ be an acyclic category with a terminal object $t$, in "Combinatorial algebraic topology" by Kozlov he defines Mobius functions for acyclic categories, he first starts by defining a function ...
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37 views

Minimum number of chains in a partially ordered set

Imagine we have a partially ordered set like $(X,R)$. So, a set like $X$ is partially ordered by a relation like $R$. We define $\delta(X):=$minimum number of chains in $X$ which covers poset My ...
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Are there any infinite (virtually) polycyclic groups with lattice orders that are not linear orders?

I am interested in noetherian group algebras, so I am learning about polycyclic groups. Specifically, I want to generalize some ideas that work well with $k[\mathbb{Z}^n]$ utilizing the lattice ...
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Can someone explain the phrase “order-preserving permutation of a totally ordered set”?

If a set is totally ordered, then any nontrivial permutation would destroy that order, so I'm clearly missing something. I have seen this exact phrase in many papers and books, mostly those dealing ...
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Does every nonempty well-ordered set have a least element?

I was making my way through The Art of Computer Programming, when some words got me puzzled. Here is how the book describes the principle of well-ordering: Let "$\prec$" be a relation on set $S$, ...
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33 views

Strictly increasing function from $\alpha$ to $\aleph_0$

Given that $\alpha < \aleph_1$, does there exists a function $f: \alpha \rightarrow \aleph_0$ that is a strictly increasing function? This seems like it would be true since $|\alpha| = \aleph_0$, ...
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Antitone Galois connections, composition

Is composition of two antitone Galois connections defined? What are all "possible" ways to define composition of two antitone Galois connections?
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Order of Galois connections between two boolean lattices

Is the poset of Galois connections between two boolean lattices itself a boolean lattice? If not, does it hold for: complete boolean lattices? atomic boolean lattices? atomistic boolean lattices?
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Galois connections between boolean lattices - an alternative representation

Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) boolean lattices (with lattice operations denoted $\sqcup$ and $\sqcap$, bottom element $\bot$ and top element $\top$). I call a boolean funcoid a ...
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Two alleged counterexamples (about boolean algebras)

Trying to solve this question, I propose two possible counter-examples. Please help me to understand whether these cases are really counter-examples. Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) ...
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17 views

Linearly extend an induced subposet

Suppose $P$ is a finite poset with partial order $\le_P$ and $Q$ an induced subposet, its partial order being ${\le_Q} = {\le_P}\cap Q^2$. Suppose we linearly extend $\le_Q$ to a linear order ...
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Boolean lattices vs boolean rings

Which kinds of theorems about boolean algebras are easier to prove with boolean rings (than with actual boolean lattices)? Give me at least one example, as an answer.
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155 views

How do you show one way equivalences in mathematics?

In the real world, it seems you can often take a set of things and create new things with them. Let's say you want to make bread, for example. Normally, you can create bread by putting together flour, ...
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48 views

Partial order is intersection of linear extensions also if poset is infinite?

I cannot understand the statement "every partial order is the intersection of its linear extensions. (See e.g. Davey and Priestley [DP]" here (page 6). The book Introduction to Lattices and Order on ...
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Why are “is equal to” and “total strict ordering” mutually exclusive?

Let $A$ be a set. Let $=$ be the relation "is equal to" on $A{\times}A$, and let $<$ be a strict total ordering on $A{\times}A$. How do we prove that if $a<b$, then it's not true that $a=b$?
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51 views

Existence of coinitial and cofinal sequences on uncontable and totally ordered set

Let be $\Omega$ an uncountable set and $\leq$ a linearly ordered relation on $\Omega$. Than can we say that exists a co-initial sequence and cofinal sequence in $\Omega$? Where we say: a subset $B$ ...
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30 views

Semicontinuous ordered topological space which is not continuous ordered

I'm reading article Mobs, trees, and fixed points by J. E. Ward. A partially ordered topological space (POTS) is defined to be a space $X$ together with a partial order $\le$ defined on $X$ such that ...
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15 views

Supremum of family of semilinear sets

Consider the class of semilinear sets. Because semilinear sets are closed under intersection and union, this class forms a lattice with inclusion as the order relation. I am interested in (infinite) ...
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141 views

Well-ordering on closed subsets of R

I don't know how to approach the proof of the following statement: If $A$ is a family of closed subsets of $\mathbb{R}$ well-ordered with respect to inclusion, then $A$ is countable. Thank you in ...
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What is the maximal chain of the following Hasse diagram?

Hi I have an exam exam this thrusday and I am struggling with the concept finding the maximal chain from the Hasse diagram. I know that a maximal chain is a chain that is not contained in the maximum ...
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55 views

Equivalence of forcing notions from dense embedding between them

In general I want to prove that $\mathbb{P}=\left(P,\leq\right)$ and $\mathbb{Q}=\left(Q,\leq\right)$ are forcing notions and there is a dense embedding $h:P\longrightarrow Q$ , then ...
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41 views

Why we define a minimal and maximal in the subset of poset?

Wikipedia said: In mathematics, especially in order theory, a maximal element of a subset $S$ of some partially ordered set (poset) is an element of $S$ that is not smaller than any other element ...
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Every well order of cardinality $\mathfrak{c}$ embeds into $(\mathcal{P}(\mathbb{R}), \subseteq)$ [duplicate]

I am trying to understand why the following statement holds true: Every well order of cardinality $\mathfrak{c}$ embeds into $(\mathcal{P}(\mathbb{R}), \subseteq)$ I'm not sure how to grasp this. My ...
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Extending well-founded relations to well-orderings

Consider the following exercise: Let $r$ be a binary relation on a set $a$. Show that $r$ is well-founded iff there exists a function $h:a\to \alpha$ for some ordinal $\alpha$, suc hthat $(x,y)\in ...
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Why can we act like functions are totally ordered by their orders?

For simplicity, consider only functions from $\Bbb N$ to $\Bbb R^{>0}$. Let $f\preceq g$ if there is an $A>0$ such that for all sufficiently large $n$, $f(n)\le A g(n)$. We normally would write ...
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43 views

A well order on a subset of a power set

I'm struggling with one problem which touches upon ordering of a power set. Assume I have a well order $\langle X,\preceq\rangle$ where $|X|=|\mathbb{R}|$. Is it possible to find (i.e. give a ...
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37 views

Countable chain condition on partial and total orders?

I'm a bit confused on the countable chain condition, as I have encountered two different definitions of the concept, and I am not sure if they are necessarily equivalent. When talking about partial ...
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Trouble understanding poset notation and Hasse Diagrams

I am trying to understand posets and if I have understood correctly - a poset is a set with a binary relation "≤" which is reflexive, transitive, and antisymmetrical. I am having trouble ...
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26 views

Subspace topology vs order topology in subspaces

For a space $X$ and a linear order $<$ on $X$, we can define a topology $\tau$ by the basic open sets $U_{a,b}=\{x\in X: a<x<b\}$. Now, let $A\subset X$. We can endow it with the subspace ...
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Complete lattices: equivalence between join and meet existence

Let $(L, \sqsubseteq)$ be a poset. In every textbook on lattice theory, you find a property of complete lattices stating that the following three are equivalent: $\forall X \subseteq L$, there ...
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42 views

The convex combination of two unequal numbers is strictly between them

$ x,y,a\; \in F $ which is an ordered field and this is given: $$ 0 \lt a \lt 1 $$ $$ x \lt y $$ I need to prove: $$ x\lt ax + (1-a)y\lt y $$ This is a relatively simple proof, yet I'm having ...
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Suppose $R$ is a partial order on $A$ and $B \subseteq A$. Prove that $R \cap (B \times B)$ a partial order on $B$.

Can somebody show me how to prove this? I would much appreciate it if one could show the givens and goals similar to how it is set out in Velleman's 'how to prove it' book, though any help would be ...
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Prove that if $(A,<)$ is a well ordering, then $(A,<)\nless(A,<)$

Prove that if $(A,<)$ is a well ordering, then $(A,<)\nless(A,<)$ I'm trying to teach myself set theory for a course I am taking and am struggling a bit here. I need to suppose for ...
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Is there a name for posets in which every nonempty finite subset with a lower bound has an infimum?

A poset in which every nonempty finite subset has an infimum is called a semilattice. I am wondering if there is a name coined for posets in which every nonempty finite lower-bounded subset has an ...
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Minimum number of comparisons sufficient to rank any set of $n$ objects?

Say you're given a set of objects, and you don't know the value of any of the objects, but, given a pair of two objects you are always told which one is bigger. For a set $\{x_1, x_2, \cdots, x_n\}$, ...
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Idempotency, non-Archimedeanness and existence of a function.

Given a structure $\mathcal{A} = (A, \succsim, \sqcup)$, where $A$ is a non-empty set, $\succsim$ is a weak order, $\sqcup$ is a commutative idempotent binary operation on $A$, let $\mu$ be an ...
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How can I prove that if $a^7 = b^7$ then $a=b$, with $ a,b \in \mathbb{Z} $

I've tried with divisibility, meaning that since $a$ divides $a^7$, then $a$ divides $b^7$ and in the same way b divides $a^7$, but I can't seem to go further than this. What properties of the ...
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Is there standard terminology for any or all of these concepts designed to help with thinking about Riemann integrability?

When trying to determine whether a function $f$ is Riemann integrable or not, we're typically in the following situation: Write $U_f(P)$ and $L_f(P)$ for the corresponding "upper Riemann sum" and ...
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Generalized ordering on simplicial complex

The vertices of simplicial complexes are usually totally ordered so that face maps of each simplex can be defined easily for the purposes of homology. That gives an "oriented" simplicial complex. But ...