Questions tagged [order-theory]
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146 questions with no upvoted or accepted answers
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A conjecture about inclusion–exclusion
$\newcommand\calF{\mathcal{F}}
\def\cupdot {\stackrel{\bullet}{\cup}}
\def\minusdot {\stackrel{\bullet}{\setminus}}$This post presents a conjecture that we have with some colleagues. It is about ...
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A Linear Order from AP Calculus
In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $...
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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.
...
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Existence of a strong antichain
Call an antichain (set of pairwise incomparable elements) $A$ of a poset $P$ strong if for every $p,q \in P$ with $p \leq q$ there exists an $a\in A$ which is comparable with both $p$ and $q$.
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Does every finite poset have a rigid endomorphism?
Crossposted on Mathematics.
In this post, an order-preserving self-map of a poset $X$ will be called an endomorphism of $X$, and such an endomorphism $f$ will be called rigid if the only automorphism ...
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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 ...
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10
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Can one define in ZFC a directed system of embeddings on the class of all linear orders realizing the surreal line as the direct limit?
Consider the surreal line $\langle\newcommand\No{\text{No}}\No,\leq\rangle$, in its order structure only. This is a proper class linear order, which is universal for all set-sized linear orders, as ...
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Computing the ordinal of a rational language well-partially-ordered by the subword relation
Let $\Sigma$ be a finite set or "alphabet", $\Sigma^*$ the free monoid on $\Sigma$ or set of "words". If $w,w'\in \Sigma^*$, write $w\leq w'$ when $w$ is a "subword" of $w'$, i.e., can be obtained by ...
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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 ...
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Order theory as a foundation of mathematics?
I know the followings kinds of formalization of mathematics:
based on set theory (e.g. ZFC)
based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example)
based on category ...
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Reference for sparseness of incomparability graphs implying sparseness of covering graphs
If a partial order on $n$ elements has $m$ incomparable pairs, then its covering graph (aka Hasse diagram aka transitive reduction, the graph of pairs of elements that are comparable but are not the ...
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Higher-order dimension in posets: a reference request
Let $P = (X, \le)$ be a partially-ordered set. Then the dimension of $P$ is the minimum number of total orders over $X$ whose intersection yields $P$. Alternately, the dimension of $P$ is the minimum ...
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Structure of $Hom(L_1,L_2)$, where $L_i$ are distributive lattices
Is there known structures/ or has there been studies on $Hom(L_1,L_2)$ of distributive lattices? Could it be made into a lattice naturally? Is there any structure on the set of ring valued functions $...
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Is there a 'local' version of Near Coherence of Filters?
The axiom Near Coherence of Filters (NCF) is known to be independent of ZFC.
Axiom (NCF I): For any two free ultrafilters $\mathcal D$ and $\mathcal E$ on $\mathbb N$, there exist finite-to-one ...
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Product of Partial Orders
Define the transpose product of a partial order $P$ over a set $S$ in the following way. The direct product of a partial order $P \subseteq S \times S$ and its converse, $P^{op}$, gives a partial ...
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Can $Ded(\kappa)$ be a supremum?
Definition If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then write $D(\kappa,\lambda)$. $Ded(\kappa)=\sup_\lambda \{D(\kappa,\lambda)\}$.
It ...
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poset of lattice properties
Is there a good overview of the dependencies between properties that a (finite) lattice poset can have?
To give a practical example, I was looking for a property weaker than congruence uniform and ...
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On thinking of spacetime as a local Scott domain
An observation of Martin and Panangaden links the study of Lorentzian manifolds and the semantics of programming languages via the theory of Scott domains.
Background:
Recall that if $M$ is a time-...
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Dimension of a union of downsets
We have established the following result regarding the Dushnik–Miller dimension of posets.
Let $P$ be a poset with downsets $C, D \subseteq P$. If the dimensions of $C$ and $D$ are $m$ and $n$, ...
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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 ...
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Natural bijection between join- and meet-irreducibles in modular lattices?
A well known property of finite modular lattices is that they have the same number of join-irreducible and meet-irreducible elements. I was wondering if there exists a natural bijection between these ...
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A representation of a partial order by a slowly changing sequence of linear orders
We study visualizations of attractors, which occur in chaotic dynamic systems, and for a few years trying to prove or refute Conjecture [3]. It has an equivalent formulation in terms of order theory, ...
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188
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Generalized graph-minor theorem?
Consider the following generalized graph-minor theorem:
GM($κ,λ$): Given any collection $S$ of $κ$ simple undirected graphs each with less than $λ$ vertices, there are distinct graphs $G,H$ in $S$ ...
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Closedness of the partial order in complete Hausdorff semitopological semilattices
First some definitions.
A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the ...
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Katetov ordering on ideals on $\omega$
Recall that a nonempty set ${\cal I}\subseteq {\cal P}(\omega)$ is a (set) ideal if
$B\in{\cal I}$ and $A\subseteq B$ imply $A\in{\cal I}$, and
$A,B \in {\cal I}$ implies $A\cup B\in {\cal I}$.
By $\...
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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 ...
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Structure of well-ordered commutative monoids
Let $(M,+)$ be a commutative monoid. Let $<$ be a well-ordering on $M$, where
$\forall a\in M,\ 0\leq a$
$\forall a,b,c\in M,\ a<b\Rightarrow a+c<b+c$
The first condition means $M$ will be ...
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Is this "trimming" of a supersolvable semimodular lattice known?
Let $L$ be a finite (upper) semimodular lattice. Recall that this means $L$ is graded and its rank function $\rho\colon L \to \mathbb{N}$ satisfies
$$ \rho(x) + \rho(y) \geq \rho(x\vee y)+\rho(x \...
- 24.2k
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Weak compactness is to trees as [?] is to lattices?
Let $\kappa$ be an inaccessible cardinal. Recall that $\kappa$ is weakly compact if every tree of height $\kappa$ has either a level of size $\kappa$ or a branch of size $\kappa$.
So if $\kappa$ is a ...
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Additional examples of classes of networks whose Hasse diagram of the poset is a perfect graph
This question is very important for my research, which is why I ask it here.
I do not have a formal background in graph theory so please excuse me if I state a term incorrectly (and feel free to ...
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Reference request: a survey of (linear) Krein-Rutman theory
I'm looking for a survey article or book chapter where a rather exhaustive treatment of the Krein-Rutman theory of positive linear operators an ordered Banach spaces is given.
Motivation. Some ...
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Self-additive posets
We say that a partially ordered set $(P,<)$ is self-additive if the two natural embeddings of $P$ in $P\oplus P$ (the linear sum of $P$ and itself) are elementary.
We have the following.
...
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(When) is the Dedekind-MacNeille completion of a po-set Hausdorff?
Let $X$ be a p.o. Consider the topology on $X$ generated by
$$U_{x}^{-}:=X\setminus (x\uparrow),\quad U_{x}^{+}:=X\setminus (x\downarrow), \quad x\in X$$
Throughout this discussion I shall refer to ...
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Characters on monotone functions
Characters on the semigroup $(C_{+}^{b}(\mathbb{R}^{d}),+)$, i.e. on bounded positive continuous functions with the ususal pointwise addition, are known to be of the form $C_{+}^{b}(\mathbb{R}^{d})\ni ...
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Face structures of chain polytopes
For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain $p_1<\ldots<...
- 3,513
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Functoriality of $\mathsf{Cu}$
I have always been happy with the proof of the functoriality of the Cuntz semigroup $\mathsf{Cu}$ given in arXiv:0902.3381, where the isomorphism
$$\mathsf{Cu}(A)\cong W(A\otimes K)$$
is used, $A$ ...
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When does $\operatorname{Aut}(M)$ preserve a linear order?
I have a general-type question:
Let $M$ be a countable structure that is ultrahomogeneous, i.e. every (partial) isomorphism between finitely generated substructures of $M$ extends to an automorphism ...
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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$...
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Can we extend "every finite lattice is a sublattice of partitions of a finite set" to linear and/or finitary lattices?
Pudlák and Tůma https://link.springer.com/article/10.1007/BF02482893 proved that every finite lattice can be
embedded as a sublattice of the partition lattice of a finite set.
Can this be generalized ...
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To whom is the classification of atomic, modular finite lattices due?
Here lattice means a poset with meets and joins. A lattice is called atomic if every element is a join of atoms. There are a few different ways to define modular for finite lattices: one is that the ...
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Are the countable (rayless) trees with wqo labels wqo?
It has been proved by Corominas that the countable trees with vertex-labels coming from a better-quasi-ordered set are better-quasi-ordered. My question is whether this holds if we replace bqo by wqo ...
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What do you call such a relation between subsets in a poset
Consider a poset $(X, \geq)$. Let's define a new relation $\succsim$ on subsets of $X$: for $A, B\subseteq X$, say $A\succsim B$ if for any $a\in A$ and any $b\in B$, we have $a\geq b$.
Does such a ...
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Cofinality without choice: can this coarse definition suffer badly?
This is a rephrased version of a question previously asked at MSE without success.
Working in $\mathsf{ZF}$, it is no longer possible in general to give every linear order an ordinal cofinality. For ...
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Obtaining linear orderings on the classical Laver tables from large cardinals
If $k:V_{\lambda}\rightarrow V_{\lambda}$ is an elementary embedding and
$R\subseteq V_{\lambda}$, then let $k^{+}(R)=\bigcup_{\alpha<\lambda}k(R\cap V_{\alpha})$.
Recall that $\mathcal{E}_{\...
4
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Can infinite bounded distibutive lattices be "arbitrarily wide"?
I was always thinking, in an informal way, that the powerset lattices ${\cal P}(X)$ (where $X$ is an infinite set) are the "widest" bounded distributive lattices with respect to their height. (In ${\...
- 48.9k
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Whitney-like embedding theorem for posets?
The Whitney embedding theorem says that any finite-dimensional smooth manifold can be embedded into $\mathbb{R}^n$ for some $n$. Is anything like this true for posets?
I'm looking for conditions on a ...
- 8,659
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153
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Maximality with respect to having no marriage
Let $A,B\neq \emptyset$ be disjoint and suppose $G = (A\cup B, E)$ is bipartite where for all $e\in E$ we have $e\cap A \neq \emptyset\neq e\cap B$. For $a\in A$ we set $N_G(a) = \{b\in B: (\exists e\...
- 48.9k
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Interval topology of the poset of all coverings
Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
- 48.9k
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Unique representability of bounded distributive lattices
Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space.
A poset $(P,\leq)$ is called (...
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What is known about orbifolding ordered groups and sets? Who has been involved? Links to Lee metrics?
In mathematical music theory several ordered groups are considered. Some examples contain the frequency space or Tonnetzes. Other groups (commutative and non-commutative ones) are discussed by Dawid ...
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