Tagged Questions

A poset or partially ordered set is a set endowed with a partial order, meaning a binary relation $\leq$ which is reflexive ($x \leq x$ for all $x$) and transitive ($x\leq y$ and $y\leq z$ implies $x \leq z$).

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154 views

Question on Posets and open sets [closed]

i'm sorry if my question is really trivial but this one is really bugging me out.. So let's have a partially ordered set $I$ with the topology in which the open sets are the increasing ones: $i\in U$ ...
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0answers
48 views

Computing basis of a lower set given basis of complementary upper set

In a poset $P$, $U\subseteq P$ is an upper set when for all $x\in U$, we have $y\ge x$ implies $y\in U$. Any subset of $P$ generates an upper set, and the basis of an upper set $U$ is the smallest ...
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3answers
263 views

Classification of countable posets?

Is there a classification of countable posets where between each two comparable elements there is a third element between them?
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85 views

Fixed points and universal maps for posets

In a recent post about f.p.p. for poset products, @M.Mirabi brought back an old-standing problem about the fixed point property of the product of two arbitrary posets which already enjoy the fixed ...
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3answers
788 views

Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorem. Suppose $P$ and $Q$ are posets ...
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2answers
84 views

a dcpo seen as a category: when does a dcpo map induce a functor with an adjoint?

Take two posets $A, B$ (partially ordered sets). Now consider these posets to be categories $Cat(A), Cat(B)$ respectively. Consider a map from $A$ to $B$, $f: A \rightarrow B$. This can be seen as ...
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2answers
162 views

Directed subposet of a poset containing the minimal elements

The following appears naturally in a certain context: Let $P$ be a graded partially ordered set. Let $M$ be the subset of minimal elements of $P$. Define subsets $E_i$ inductively as follows: First, ...
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1answer
217 views

The category of categories and adjunctions

What is known about the category that has small categories as objects and adjunctions as morphisms? Obviously, it has neither terminal nor initial objects. But what about other kinds of limits? Are ...
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1answer
155 views

Principal Order Ideals in the Weak Bruhat Order

Let $\sigma\in S_n$ be a permutation on $n$ elements, and $\mathrm{Inv}(\sigma):=\{(i,j) : 1\leq i<j\leq n\text{ and }\sigma(i)>\sigma(j)\}$ be its set of inversions. In the weak order on ...
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1answer
88 views

Krull dimension of dense extensions

Let $A$ be a boolean algebra and let $B\leq A$ be a boolean sub-algebra which is dense (for all $0\neq a\in A$, there is a $0\neq b\in B$ such that $b\leq a$). We suppose also that $B$, as a partially ...
2
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1answer
121 views

Galois Connections: algorithmic generation

Given two finite posets $P,Q$, is it known any algorithm to count and/or generate every Galois Connection between $P$ and $Q$ ? I'm looking for references about this problem.
2
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1answer
146 views

Maximal number of antichains of a connected poset

Assume we have a connected poset $P$ of $n$ elements, I am searching to know what is the maximal number of antichains such a poset can have? $2^n$ is obviously an upper bound, and my feeling is that ...
3
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1answer
115 views

Matching in the Boolean Algebra

We consider the Boolean Algebra of the set $[n]=\{1,\dots,n\}$ and the matching $\psi$ from the $m+1$ subsets of $[n]$ into the $m$ subsets of $[n]$ for $(n+1)/2\leq m+1\leq n$, which is used to show ...
6
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1answer
415 views

Order homomorphism functions on $\omega_1$

Let $\omega_1$ be the first uncountable ordinal, same as the set of all countable ordinals. Let $F$ be the set of all functions $f$ from $\omega_1$ minus singleton $0$ into $\omega_1$ that are (a) ...
4
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1answer
175 views

About subposet of Levy collapse

Let $\lambda>\kappa$ and $\operatorname{Coll}(\kappa, \lambda)$ be the poset collapsing $\lambda$ to $\kappa$. Pick a subposet $P$ which is $\lt\kappa$-closed and of size $\lambda$. Can we say that ...
2
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2answers
103 views

Representation of Banach spaces partially ordered by solid, normal, minihedral cones

I've been using the representation result below, from Krasnosel'skij/Lifshits/Sobolev; Positive Linear Systems---The Method of Positive Linear Operators. Heldermann Verlag, 1989. Theorem. Let $E$ be ...
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0answers
234 views

Is the Poset of Graphs Automorphism-free?

For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$. ...
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66 views

Suprema and infima in spaces ordered by non-normal cones

Background We shall call a subset $V_+ \subseteq V$ of a Banach space $V$ a cone if $V_+$ is closed, $\alpha V_+ \subseteq V_+$ for all $\alpha \geqslant 0$, and $V_+ \cap (-V_+) = \{0\}$. Cones ...
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143 views

Strength of claims about extensions of partial preorders and orders to linear ones

Consider these two axioms: Every partial order extends to a linear order. Every partial preorder (reflexive and transitive relation) extends to a linear preorder while preserving strict orderings: ...
2
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0answers
61 views

Rank generating functions on a graded poset and a linear extension of it

Let $A$ be a possibly infinite ensemble and let $\leq$ be a partial order between elements of $A$. We denote $P_{\leq}=(A,\leq)$ the corresponding poset. Furthermore, we suppose that $P$ is graded, ...
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1answer
84 views

Generalized connected components decomposition for Priestley spaces

Preliminaries A partially ordered space is both a poset and a topological space. It has connected components both as a topological space, and connected components as a poset, i.e. the maximal ...
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1answer
87 views

Are the connected components of a Priestley space closed?

Preliminaries A Priestley space is both a poset and a topological space. The topologically connected components of the space are trivially closed. (They are just the points of the underlying set.) But ...
2
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1answer
132 views

Linear order extensions on (nonabelian) groups

If $G$ is a group with a (left) linear order, does every (left) partial order on $G$ extend to a (left) linear order? The answer is affirmative on abelian groups, where being torsion-free is ...
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1answer
203 views

Terminology question for poset maps

Is there a standard name for order-preserving maps $f\colon P\to Q$ of posets with the property that the image of a lower set is a lower set, or equivalently if $q\leq f(p)$ then there exists $p'\leq ...
7
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1answer
259 views

Does this lattice have a name (and literature)?

The "lattice" in the title appears to be a lattice. At least it's a poset, which I define now. Fix a partition $\lambda$ of $n$ and consider the set of all standard Young tableaux (each of ...
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2answers
321 views

Do operations generate well-ordered sets only?

I've read   @TauMu's question   about the set of functions   $\mathbb N\rightarrow\mathbb N$   generated from the identity map by repeatedly applying exponentiation of two already ...
9
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1answer
422 views

Does this property of a partially ordered set have a name?

What do you call a poset with this property? For any elements $a,b,c,d$ such that $\{a,b\}\le\{c,d\}$, there is an element x such that $\{a,b\}\le x\le\{c,d\}$. (Equivalently, for any finite sets ...
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0answers
137 views

“Double convolution” with the Mobius function on a poset

Let $f$ and $g$ be arbitrary (say integer-valued) functions on some poset $P$, and say $\mu$ is the Mobius function of $P$. I'm studying a quantity that's a sort of "double convolution" of $f$ and $g$ ...
9
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2answers
289 views

How exactly does Schützenberger promotion relate to Striker-Williams promotion?

Schützenberger promotion, studied (for example) in Richard Stanley, Promotion and Evacuation, 2009, is a permutation of the set of all linear extensions of a finite poset. Since one can identify the ...
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2answers
1k views

Does this poset have a unique minimal element?

Recently I have been thinking about the following poset: the underlying set is $\mathcal{AFT}$ consisting of all (finite) automorphism-free undirected trees (with at least one edge to exclude the ...
7
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1answer
243 views

Which of these relations on partial orders allows us to identify forcing equivalence?

Background This question was inspired by Justin Palumbo's excellent question Cantor Bernstein for notions of forcing. In his question, Justin considers a relation $\lhd$ on partial orders (defined ...
4
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0answers
159 views

When can we determine an $f$-vector or rank-generating function from its unordered list of coefficients?

Let $f_i$ be the number of $i$-dimensional faces in a $d$-dimensional simplicial complex $\Delta $. Recall that the $f$-vector of $\Delta $ is the vector $(f_{-1},f_0,f_1,\dots ,f_d)$ where ...
2
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4answers
384 views

Minimal (semi)lattice containing a given poset

For a given poset, (I think that) it is easy to construct the minimal join-semilattice containing that poset. I wonder whether the minimal lattice containing that poset is also easy to construct. I ...
3
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0answers
127 views

references for properties/examples of breadth in (semi)lattices

This is in some sense following up on my earlier question and the answer given by NN. I am currently revising the paper which used the condition mentioned in my question. It was pointed out in NN's ...
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1answer
307 views

When is the derived category of representations of a finite poset equivalent to its opposite?

If I have a finite partially ordered set $K$, I can look at its derived category of finite dimensional representations $D(K)$. Note that $D(K^{op}) \simeq D(K)^{op}$ by linear duality. But when do ...
2
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2answers
312 views

Characterizing Posets by Functions Into Natural Numbers

Let $P$ be a poset and denote by $Hom(P, \mathbb N)$ the set of all monotone functions from $P$ to natural numbers $\mathbb N$. Under what conditions on $P$ Is it possible to recover the order on $P$ ...
14
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1answer
513 views

Bruhat order and the Robinson-Schensted correspondence

The Robinson-Schensted correspondence is a bijection between elements of the symmetric group $S_n$ and pairs of standard tableaux of the same shape. The symmetric group is partially ordered by the ...
3
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1answer
258 views

Grading a non-graded poset as squeezed as possible

Here is a curiosity question (motivated by the recent revamp of ranked-poset routines in Sage). Let $P$ be a finite poset. We look for a family $\left(a_p\right)_{p\in P}$ of real numbers summing up ...
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6answers
2k views

The category of posets

I am trying to teach myself category theory and, as a begginer, I am looking for examples that I have a hands-on experience with. Almost every introductory text in category theory contains following ...
10
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0answers
352 views

Asymptotic growth of antichains in divisibility posets

The following question is inspired by a problem that Erdős used to ask epsilons. It asks to prove that if one chooses a subset of $\lbrace 1,\dots,n\rbrace$ with more than ...
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3answers
776 views

Well-ordered cofinal subsets [closed]

Let $(P, \leq)$ be a total ordering (some of you prefer the name linear order). Can we find a subset $R\subseteq P$ which is well ordered (with respect to $\leq\upharpoonright R$) and cofinal in $P$, ...
2
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1answer
269 views

What are $n$-poset?

Yesterday I was wandering for the $n$-lab and I've found the definition of $n$-poset. Following this post it seems that a $n$-poset should be a $(n,n+1)$-category. Now an $(n,r)$-category should be a ...
10
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1answer
676 views

What is the size of a largest antichain in this poset?

Let $[n]:=\lbrace 1, \dots, n \rbrace$. We define a partial ordering on the set of subsets of $[n]$ as follows. We say that $X \preceq Y$ if there is an injective map $f:X \to Y$ such that $x \leq ...
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0answers
105 views

A search for optimal order ideals

At the behest of Gerhard Paseman I'll describe the problem that I alluded to in name for a partial order. Let $M = M(\infty)$ denote the set of all finite subsets of the positive integers ...
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407 views

The name for a partial order

In a recent paper of mine, my co-authors found that a partial order that we were using was contained in a paper by Kundgen. In it, he called it "right-shifted partial order". I was curious, and found ...
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1answer
878 views

What is the number of maximal antichain in a poset?

This is a topic I am recently working on. Given a poset, how many different antichains are there? I find little literature on it. And I am interested whether there is a closed formula, or a tight ...
8
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1answer
439 views

Ordered sum of posets

Let $I$ be a poset and for any $i$ let $P_i$ be a poset. Let $P$ be the sum over $I$ of the sets $P_i$, and let $<_P$ be the relation defined on $P$ by $q<_Pr$ iff $q$ and $r$ are members of the ...
6
votes
1answer
360 views

Posets of cosets and contractibility

For this question let $G$ be a group, perhaps infinite, and let $H_i$ for $i\in I$ be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset ...
3
votes
1answer
279 views

A compactness property of posets

Consider a poset $P$ and suppose that every finite subset admits a supremum. Call an ideal $I$ of $P$ minimal infinite if it is infinite and every ideal properly contained in $I$ is finite. I am ...
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173 views

Ordering labellings of a fixed poset.

Let $\{A_1,\ldots, A_m\}$ be a family of sets and $I=\{1, \ldots, m\}$. Assume for any $J\subset I$, $B_J=\bigcap_{i\in J}A_j$ satisfies $1\leq |B_J| \leq m-1$ as long as $|J|>1$. We define a ...