Questions tagged [order-theory]
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645
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Is every finite poset a subset of a finite complemented distributive lattice?
Let $(X,\succeq)$ be a poset. I have the following two questions:
Is it true that there exists a finite complemented distributive lattice (a Boolean lattice) $(S, \succeq^*)$ such that $X\subseteq S$ ...
3
votes
0
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123
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A class of Kripke frames which preserves validity
The background of our discussion is intuitionistic logic, i.e. the following definitions are intuitionistic Kripke frame.
For $1\leq s\leq n-2$, the frame $\mathcal{C}_n(s)$ denotes the frame which is ...
14
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1
answer
582
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On certain order-automorphisms of the rationals
Consider the rationals $\mathbb{Q}$ with the usual order $\leq$. Now let $A$ be a subset of $\mathbb{Q}$, such that foreseen with the induced order $\leq$, $(A,\leq)$ is a dense linear order.
...
5
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0
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125
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Which monomials are "leadable"?
Question: Let $k$ be a field, let $f \in k[t_1,\ldots,t_N]$ be a nonzero polynomial. Which monomials
$m_a = t_1^{a_1} \cdots t_N^{a_n}$ appearing in $f$ are leadable in the sense that they are the ...
2
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0
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Can this order relation, defined in terms of all topological spaces, be defined in terms of the reals alone?
Let $K$ be the operator monoid under composition of Kuratowski's $14$ set operators generated by topological closure $k$ and complement $c.$ Kuratowski's 1922 paper gives the poset diagram of the ...
1
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0
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90
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What computable pseudo-ordinals are there with initial segment $\omega_1^{CK}(1+\eta+1)$?
The notion of a “computable pseudo-ordinal”, i.e. a computable linear order with no hyperarithmetical descending chains, is an old one going back to Stephen Kleene. Joe Harrison wrote the definitive ...
9
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0
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291
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Mapping graphs to ordinals
Robertson-Seymour theorem implies that graph minor relation is a well-quasi-ordering, which means (among other things) that this relation can be extended to a well-order, and other result says that ...
1
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1
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193
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does this relation associated with a poset have a name?
Given a partial order $P$ on a set $S$ does the set of ordered pairs $(x,y)$ in $S\times S\setminus P$ such that $P\cup\{(x,y)\}$ is a partial order have a name? (If so then it would apply to all ...
3
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0
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What are all the order types of maximal chains of $\Delta^0_2$ sets?
A set of natural numbers is $\Delta^0_2$ if it’s computable from the halting set. Consider the quasi-order/pre-order of all $\Delta_0^2$ sets ordered by $m$-reduction, or equivalently consider the ...
10
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Let $X$ be a finite set of $n$ ($>1$) elements and $\tau$ be a topology on $X$ having exactly $m$ elements. Can we give any description of $m$?
Let $X$ be a finite set of $n$ ($>1$) elements and $\tau$ be a topology on $X$ having exactly $m$ elements.
Can we give any description of $m$ as it relates to $n$?
Obviously $2\le m\le 2^n$ and ...
2
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0
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79
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Ordered vector space that can be embedded into its bidual
We say that an ordered vector space $(V, \ge)$ (over $\mathbb{R}$) is "bidual embeddable" (I made up this name, not sure whether this concept already exists) if for every $x \in V$, if $x$ ...
6
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1
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349
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Is every homogeneous poset a lattice?
A poset $(P,\leq)$ is homogeneous if $P\cong [a,b]$ for all $a,b\in P$ with $a<b$ (where $[a,b] := \{x\in P: a\leq x\leq b\}$).
Examples of homogeneous posets include $[0,1]$, $[0,1]\cap \mathbb{Q}$...
4
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1
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Is ${\cal P}(\omega)/\text{(fin)}$ a fractal poset?
If $(P,\leq)$ is a partially ordered set and $a,b\in P$ we set $[a,b]:=\{x\in P: a\leq x\leq b\}$. We say that $P$ is fractal if whenever $a,b\in P$ and $[a,b]$ contains more than one element, then $[...
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Order-embeddability of ${\frak b}$ and ${\frak d}$ in $\mathbb{R}$ [duplicate]
The starting point of this question is the observation that in ${\sf (ZFC)}$, all ordinals $\alpha < \omega_1$ can be order-embedded in $\mathbb{R}$.
Let $\omega^\omega$ denote the set of all ...
1
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0
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Causal-net category and poset category
Order is a fundamental mathematical structure. There are two natural ways to represent order structures, by posets and by causal-nets (acyclic directed graph). How can we compare these two ways, and ...
1
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1
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Characterization of edge posets
Given an acyclic directed graph $G$, the set $E(G)$ of edges of $G$ equipped with the reachable order $\to$ is called the edge poset of $G$, where for two edges $e_1\to e_2$ means that there is a ...
11
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1
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345
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Synthetic differential / conformal geometry of Lorentzian manifolds?
Let $M$ be a sufficiently nice Lorentzian manifold of dimension $\geq 3$. It's known [1] (see also [2]) that the differential and even conformal structure of $M$ is completely encoded in the causal ...
2
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0
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Countable highly order-transitive subgroups of $\mathrm{Aut}(\mathbb{Q},\leq)$
Consider $A := \mathrm{Aut}(\mathbb{Q},\leq)$, the group of order-automorphisms of $(\mathbb{Q},\leq)$. Call a subgroup $U$ highly order-transitive if for any two finite ordered sequences $s_1$ and $...
8
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3
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544
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closure of separative quotients
Does there exist a partial order, nontrivial for forcing, that is countably closed, but whose separative quotient is not countably closed? Supposing the answer is yes, then is there a partial order, ...
11
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1
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646
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Do all toposes satisfy the internal Zorn's lemma?
I came up with this question when trying to give a more detailed answer to a question by Tim Campion in a comment to Ingo Blechschmidt's answer to Examples of statements that are valid in every ...
8
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2
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How this set of functions is ordered?
Notation:
$k, m, n$ are non-negative integers
$f, g, h$ are functions $\mathbb{N} \to \mathbb{N}$
$f^k$ is $k$-th iterate of the function $f$: $f^0(n)=n, f^{k+1}(n)=f^k(f(n))$
$f \prec g$ means ...
2
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1
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Non-cofiltered derived limits
As far as I know, the inverse limit and its derived functors can be defined even in case we are dealing with a functor $F: I \to A$ from a category $I$ that is not cofiltered. I would content myself ...
2
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0
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62
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Total orders on subsets
Let $X$ be some finite ground set. Let $\prec$ be a total order on the powerset $\mathcal{P}(X)$, such that if $A\prec A’,B\preceq B’$ and $A\cap B= A’ \cap B’ = \emptyset$, then $A \cup B \prec A’ \...
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1
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Why do we need "canonical" well orders?
(I asked this question on Math.SE earlier but received no response and am therefore moving it here, please note that I realise this question is probably incredibly naïve for the experienced set-...
2
votes
1
answer
77
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Request for literature recommendations on isotonic mappings
An isotonic mapping is a function between two partially ordered sets that preserves the ordering between the elements. Specifically, given two partially ordered sets $(X,\le)$ and $(Y,\le)$, a ...
1
vote
1
answer
184
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Lower bound on ratio of extreme order statistics
This question relates to bounds on expectations of order statistics, elaborated upon in the Book by Arnold and Balakrishnan (1989). Let $X_1,\ldots,X_n$ be i.i.d. continuous random variables ...
1
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1
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Thinning directed sets ${\frak P}$ of partitions of $\omega$ with no ${\frak P}$-discrete subsets
This question branches from Taras Banakh's recent question on a cardinal characteristic connected to families of partitions that are directed in the ordering of partition refinement.
A partition $\...
2
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2
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335
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Algorithm to compute certain poset from a given poset.
Hi. Associated with a finite poset $P$, one can consider the poset $S(P)$, whose elements are the intervals of $P$, ordered by inclusion. (See Discrete version of Nullstellensatz? for some motivation ...
5
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2
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Do germs of open sets around a point form a frame?
Let $X$ be a topological space and $x \in X$ a point. Let $\Omega$ be the set of open sets (viꝫ. the topology) of $X$, and $\Omega_x$ the set of germs around $x$ of open sets, that is, $\Omega_x = \...
8
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1
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Example of trickiness of finite lattice representation problem?
I'm trying to come up with a good explanation for my students of why the finite lattice representation problem is difficult. I've already shown that the "greedy approach" to representing the ...
6
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1
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245
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Poset as union of posets of lower cofinality
Let $ \mathbb{P}$ be any directed, well-founded poset of cofinality $ \aleph_{n+1}$, where $n$ is a natural.
Can we write it as an increasing union $ \mathbb{P} = \bigcup_{\alpha < \omega_{n+1} } \...
3
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0
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127
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Is there an ordered algebra analogue of the HSP theorem?
For an algebraic signature (= set of function symbols) $\Sigma$, say that an ordered $\Sigma$-algebra is a pair $\mathfrak{A}=(\mathcal{A};\le)$ where $\mathcal{A}$ is a $\Sigma$-algebra in the sense ...
2
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0
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$\sigma$-fields as closure systems
Let $(\Omega,\mathcal A, P)$ be a probability space and let $\Sigma(\mathcal A) \subset 2^{\mathcal A}$ be the collection of all sub-$\sigma$-fields of $\mathcal A$. Then, $\Sigma(\mathcal A)$ is ...
4
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1
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Exactly how much (and how little) can partial ordered sets (classes) embed to the cardinalities
In the paper "Convex Sets of Cardinals", Truss mentioned a result of Jech:
If $M$ is a countable transitive model of ZFC, and $(P,<)∈M$ is a poset, then there exists a Cohen extension of ...
2
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1
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Complete De Morgan algebra
Recall that an algebra $(A,\sim)$ is a De Morgan algebra if $A$ is a bounded distributive lattice and $\sim$ is a unary operation which satisfies:
${\sim} (x\vee y)={\sim} x\wedge {\sim} y$ and ${\sim\...
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Weak compactness is to trees as [?] is to lattices?
Let $\kappa$ be an inaccessible cardinal. Recall that $\kappa$ is weakly compact if every tree of height $\kappa$ has either a level of size $\kappa$ or a branch of size $\kappa$.
So if $\kappa$ is a ...
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What is the structure of a space of $\sigma$-algebras?
Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
5
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0
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615
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A poset with small "cycles"
(A followup to this recent question.)
I noticed the following curious property of a poset (which I strongly believe to be a lattice, I'm still trying to prove that…):
Suppose that $z$ is covered by $x$...
8
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1
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A characterization of the poset of filters on a set
For the lattices of all subsets of a given set, an axiomatic characterization is known: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra.
The ...
28
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1
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What is the cofinality of the co-infinite subsets of ${\bf N}$?
Let ${\mathcal A}$ be the family of subsets $A$ of the natural numbers ${\mathbf N}$ which are co-infinite (i.e., their complement is infinite). We partially order this family by set inclusion. A ...
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What's a non-abelian totally ordered group?
Because I have heard the phrase "totally ordered abelian group", I imagine there should be non-abelian ones. By this I mean a group with a total ordering (not to be confused with a well-ordering) ...
6
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2
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Embedding a Brouwerian lattice into a Boolean lattice
I have already asked a similar question at
https://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra
but have received no answer.
Sorry, I ask a ...
6
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1
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Smallest ordinal $\mu$ not embeddable in ${\cal P}(\omega)$
The motivation for this question is the startling fact that there is an order-preserving injective map (embedding) from $\mathbb{R}$ into ${\cal P}(\omega)$. (Think Dedekind cuts.) I am wondering how &...
0
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0
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205
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Vector-valued valuations on lattices
There's a fair amount of work on valuations on (modular) lattices, by which I mean functions $v : \mathcal{L} \rightarrow R$ that satisfy the modular expression
$$v(x) + v(y) = v(x \wedge y) + v(x \...
2
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1
answer
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Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational
Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius ...
2
votes
1
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290
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Uniqueness of minimal completions of a partially ordered set
The partially ordered set $(Y,\le)$ is called a superspace of the partially ordered set $(X,\le')$ iff
$X\subseteq Y$.
$\le' \:\:=\: (\le\cap X^2)$.
and a completion of $(X,\le')$ if in addition
$~~ ...
5
votes
1
answer
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Scott topology: Suprema of sequences are topological limits
I read that, with the Scott topology, suprema of sequences are topological limits (See page 1 of this article).
Let $(X, \le)$ be a DCPO, and $D$ be a directed subset of $X$.
I can easily see that the ...
7
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2
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776
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Counterexample on completely distributive lattices
I would like to see an example of a complete lattice $C$ which is both a frame and a dual-frame, i.e. finite meets distribute over arbitrary joins and finite joins distribute over arbitrary meets (...
4
votes
1
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139
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Existence of more distributive Boolean lattices
Is there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that
$$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in I}a_{if(...
6
votes
1
answer
371
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Poset of automorphism groups of variants of a structure
Say that two structures $\mathfrak{A},\mathfrak{B}$ in possibly different finite languages are parametrically equivalent - and write "$\mathfrak{A}\approx\mathfrak{B}$" - iff they have the ...