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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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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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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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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42 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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34 views

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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48 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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28 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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140 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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19 views

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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49 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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39 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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38 views

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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24 views

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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42 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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32 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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44 views

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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20 views

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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40 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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62 views

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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39 views

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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49 views

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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21 views

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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66 views

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 ...
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65 views

Order-Preserving Adjunction?

Let there be two categories PrO (category of preordered sets) and ProM (category of preordered monoids). If we know that there exists an adjunction $F\dashv G$ for functors ...
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77 views

Why does $a≤b{\implies}a=b∧a<b$?

Let $≤$ be a relation on set $A$ such that $≤$ is reflexive, transitive and antisymmetric. Let $<$ be a relation on set $A$ such that $<$ is asymmetric, antireflexive and transitive. Let $a$ ...
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Sets which are order-isomorphic to the (extended) rationals

Let $(S,<)$ be a totaly ordered set under the strict order relation $<$. Suppose that, for any $a,b\in S$, if $a<b$, then there exists $c\in S$ such that $a<c<b$. We also assume that ...
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57 views

The set of rational numbers has continuum many subsets that are order-wise inequivalent

How to show that there are continuum many subsets of $\mathbb Q$, no two of which are similar? Two sets are called similar if there is an order-preserving bijection between them.
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Why is the infimum of the following relation 1?

The question was: What is the infimum from $K=\{x ~ |~0 \leq x \lt 1\}$ for the partial order $\{ (x,y) ~ | ~ ( \exists z \in \mathbb{R}_{\geq 0})[x-z=y] \}$ on set $\mathbb{R}$. This was a question ...
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Why does the Möbius function take its values so often in $\{0,+1,-1\}$?

The Möbius function of a locally finite poset $P$ is defined on its intervals $[x,y] \subseteq P$ recursively by $$\mu([x,x])=1$$ $$\forall x < y : \mu([x,y])=-\sum_{x \leq z < y} \mu([x,z])$$ ...
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Ordering on the reals for which every element has a successor and a predecessor

I'm going through Problem and Theorems in Classical Set Theory by Komjath and Totik, and the answer to this question is: Let $[x]$ (resp. $\{x\}$) denote the integral (resp. fractional) part of $x$, ...
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16 views

Suborderable space, orderable characterization proof doubt

In Orderability in the presence of local compactness, Valentin Gutev states and proves the following proposition: A suborderable space $X$ is orderable with respect to a linear order $\prec$ on it if ...
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191 views

What is so special about Higman's Lemma?

Is there a motivational example of an application of Higman's Lemma that brings out the true beauty and importance of Higman's Lemma? What is the thing that made so many people care about it? For an ...
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Can we associate values to elements in a poset? [duplicate]

My question comes from personal investigation. Suppose you have a poset $(X, \le_X)$. I would like to associate to all elements $x \in X$ a value $v(x) \in V$, where $(V, \le_V)$ is a totally ordered ...
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subobject classifier for partial orders

Does the category of partial orders have a subobject classifier? (Edit: No, see Eric's answer.) If not, what is a category which is "close" to the category of partial orders (e.g. it should consists ...
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Isomorphisms of well ordered sets

Let $A$, $B$ be well ordered sets. If there exist order-preserving functions $f : A \to B$ and $g : B \to A$ (do not need to preserve initial segments), are $A$ and $B$ isomorphic? I know this is not ...
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Find $A$ and $B$ such that $A⊈B$ and $B⊈A$? [closed]

I need to prove that the subset relation “$⊆$” on all subsets of $\mathbb Z$ is not a total order and I'm going to do this by finding $A$ and $B$ such that $A⊈B$ and $B⊈A$? Is there a simple solution ...
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Formally real field with two different orders

If $F$ is a formally real field, then there exists a total order relation on $F$ which is compatible with its sum and product, but it needs not be unique. a) How different can be two (compatible with ...
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Group orderable iff all its finitely-generated subgroups are orderable

I want to proof this specifically using the Compactness Theorem from propositional logic (this is an exercise from Model Theory, Hodges). $G$ orderable means there is a total ordering s. t. $g\leq h$ ...
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55 views

Subset relation ⊆ on all subsets of ℤ is a partial order, not a total order.

I need to prove that the subset relation “⊆” on all subsets of ℤ is a partial order but not a total order. I'm not experienced in these kind of proofs and was hoping to see an example of an easier one ...
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35 views

Proving that divisibility in an integral domain is a partial ordering

Given that R is an integral domain. I'm trying to prove that divisibility on this set constitutes a partial ordering. In particular, I have defined the relation $y \leq_{\,d} x$ on R by $y|x$. ...
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Why is the symbol for the exterior product a meet rather than a join?

I've moved this over to HSM. It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called ...
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Turning Preordered Sets into Preordered Monoids (Constructing Preordered Monoids from Preordered Sets)

Question: Referring to the Wikipedia article on Adjoint Functors in Section 2 (Motivation), they talk about "turning rngs into rings" (can be rephrased as "constructing rings from rngs"). I do not ...