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 $x$), antisymmetric ($x\leq y$ and $y\leq x$ implies $x=y$), and transitive ($x\leq y$ and $y\leq z$ implies $x \leq z$).

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60
votes
6answers
14k views

How many orders of infinity are there?

Define a growth function to be a monotone increasing function $F: {\bf N} \to {\bf N}$, thus for instance $n \mapsto n^2$, $n \mapsto 2^n$, $n \mapsto 2^{2^n}$ are examples of growth functions. Let's ...
34
votes
3answers
1k 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 ...
22
votes
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 ...
19
votes
0answers
312 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 $H$. ...
17
votes
6answers
3k 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 ...
17
votes
1answer
679 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 ...
16
votes
3answers
1k views

Banach and Knaster-Tarski fixed point theorems — are they related?

It there any known way of obtaining the Banach fixed-point theorem from the Tarski fixed-point theorem or vice-versa?
15
votes
3answers
4k views

Proving that a poset is a lattice

I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably ...
15
votes
3answers
896 views

Subposets of small Dushnik-Miller dimension

The Dushnik–Miller dimension of a partial order $(P,{\leq})$ is the smallest possible size $d$ for a family ${\leq_1},\ldots,{\leq_d}$ of total orderings of $P$ whose intersection is ${\leq}$, i....
12
votes
1answer
329 views

Fastest algorithm to compute the width of a poset

An colleague recently came to me with a problem concerning the scheduling of tasks in the presence of constraints (of the kind: task $x$ can't begin until task $y$ has been completed). It turned out ...
11
votes
1answer
258 views

Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property

What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis. $\mathbb{P}$ preserves stationary subsets ...
11
votes
0answers
397 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 $\lfloor\frac{n+1}{2}\rfloor$...
10
votes
3answers
449 views

Is a distributive lattice planar iff it admits no B3 sublattice?

A finite lattice is planar if it admits a Hasse diagram which is a planar graph (we want the Hasse diagram to be represented in the plane so that the $y$-coordinate of each element respects the order)...
10
votes
2answers
379 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 ...
10
votes
1answer
816 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 ...
10
votes
2answers
321 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 ...
10
votes
1answer
346 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 ...
9
votes
3answers
239 views

Is the homomorphism poset directed if the codomain is directed?

Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. (We call a function $f:P\to Q$ order preserving if $x\leq y$ in $P$ ...
9
votes
1answer
172 views

Are these two quotients of $\omega^\omega$ isomorphic?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say $f\simeq_{\text{fin}} g$ if there is $n\in \omega$ such that $f(k) = g(k)$ for all $k\geq n$. ...
9
votes
2answers
480 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 ...
9
votes
1answer
449 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 $A\...
9
votes
1answer
337 views

Set of maximal subfields not containing particular elements.

Instead of extending a field, by adjoining a new element, consider what happens if we remove an element or elements. This started as a question on math.SE Field reductions where Pete L. Clark ...
9
votes
1answer
566 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) ...
9
votes
0answers
83 views

Ideals in strong Bruhat order

Recall that a (lower) ideal in a poset $(P, \le)$ is a subset $I\subset P$ such that $x\in I \Rightarrow y\in I$ for all $y\le x$. In am interested in ideals in finite Coxeter groups $W$ equipped ...
9
votes
0answers
121 views

A family of posets

Consider the family of all (finite) posets that can be obtained by repeatedly applying one of the following three operations (starting e.g. with the empty poset): (O1) Disjoint union of one or more ...
8
votes
2answers
647 views

Statements forced by one condition of a poset, but not the whole thing

In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be ...
8
votes
2answers
300 views

Posets isomorphic to their endomorphism poset

Let $(P,\leq)$ be a poset. We set $$\text{End}(P)=\{f: P\to P: f\text{ is order-preserving}\}$$ and order $\text{End}(P)$ pointwise. Is there a poset with more than 1 point such that $P\cong \text{...
8
votes
1answer
343 views

Decomposing posets into countably many chains

A conjecture of Galvin's is that the following is possible (which I take to mean that the consistency of the following can be proven relative to the consistency of something like ZFC, or ZFC plus some ...
8
votes
1answer
147 views

Optimal strategy for game of 'online sorting' into a poset

Consider a single-player game played with an arbitrary finite poset, and a random number generator with a known distribution: Each turn, the RNG produces a number, and the player must assign that ...
8
votes
1answer
173 views

Extending subsets to supersets in different ways

We are given a collection of sets $A_1,\ldots,A_s$, pairwise different and each of cardinality $k$, and a collection of sets $B_1,\ldots,B_s$, pairwise different and each of cardinality $l>k+1$, ...
8
votes
1answer
513 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 ...
8
votes
0answers
168 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 ...
7
votes
3answers
718 views

Characterizing forcings that don't add any dominating reals

Regarding reals as functions from $\omega$ to $\omega$, let's say a real $f$ eventually dominates $g$ iff $(\exists n)(\forall m > n)[ f(m) > g(m)]$. Let's say that a (non-trivial separative) ...
7
votes
2answers
425 views

A sequence of generic filters that does not come from an iteration

Fix a countable transitive model $M$ of ZFC. In my answer to this question I indicated that there are forcing iterations $((Q_\alpha:\alpha\leq\omega),(\dot P_\alpha:\alpha<\omega))$ in $M$ and ...
7
votes
2answers
887 views

Is every poset the poset of prime ideals of a ring?

The answer to this question, as it is, is trivially false, for one necessary condition is the existence of maximal element(s), i.e., maximal ideals exist and are prime. My question was inspired from ...
7
votes
1answer
282 views

Unique factorization of posets

Given two finite posets $P$ and $Q$, we can form the direct product poset $P \times Q$ whose elements are pairs $(p,q) \in P \times Q$ with $(p,q) \leq (p',q')$ if $p \leq p'$ and $q \leq q'$. Let us ...
7
votes
1answer
200 views

Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound. Question: Suppose $\mathbb{P}$ is a separative partial order which is $\...
7
votes
1answer
369 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 $1,\dots,n$...
7
votes
1answer
387 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 $\mathcal{C}(...
7
votes
1answer
310 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 ...
7
votes
1answer
541 views

Does ⋄ generate all De Morgan algebras?

(Asked by Nathaniel Hellerstein on the Q&A board at JMM) This question is about De Morgan algebras (see also Wikipedia), which are something like Boolean algebras, but with a different weaker ...
7
votes
1answer
284 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 ...
7
votes
0answers
237 views

Pattern Avoidance in Poset Permutations

I am not sure if it is appropriate to use MathOverflow to publicize a conjecture, but I think this is an interesting question and I have no real ideas of how to solve it. A permutation on a $n$-...
7
votes
0answers
169 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: i....
7
votes
0answers
490 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 ...
6
votes
1answer
234 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 ...
6
votes
2answers
261 views

Is this algebra isomorphic to an incidence algebra?

This question is motivated by trying to establish a converse to Theorem 7.8 of our paper. I have a finite poset $P$ with the following properties: $P$ has binary meets (and hence a least element). $...
6
votes
0answers
166 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 $g$ ...
5
votes
5answers
963 views

Category = Groupoid x Poset?

Is it possible to split a given category $C$ up into its groupoid of isomorphisms and a category that resembles a poset? "Splitting up" should be that $C$ can be expressed as some kind of extension ...
5
votes
3answers
1k views

Constructing a metric over a lattice

Consider a lattice $({\cal L}, \wedge, \vee)$ with an antimonotonic function $f: {\cal L} \rightarrow {\mathbb R}$ defined on it (i.e $x \preceq y \implies f(x) \ge f(y)$). $f$ is said to be ...