**1**

vote

**1**answer

897 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**

votes

**1**answer

441 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

**1**answer

364 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

**1**answer

280 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 ...

**0**

votes

**0**answers

174 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 ...

**6**

votes

**2**answers

808 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 ...

**5**

votes

**1**answer

336 views

### Minimum set cardinality for a given partially ordered set

Let S be a finite set of cardinality k. I consider subsets of S that I order by set inclusion. For any given k, this defines the partially ordered set S_k.
To a given partially ordered set P, I ...

**8**

votes

**2**answers

528 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 ...

**4**

votes

**1**answer

323 views

### On the barycentric subdivision of a poset

Hi everybody,
I'm interested in the barycentric subdivision of a poset $P$, defined as the face poset of the corresponding order complex $\mathcal F(\Delta(P))$ (or alternatively, the poset of chains ...

**3**

votes

**1**answer

161 views

### Decomposing a poset into directed subposets

Let us say that a poset $P$ is $\mathbf{\kappa}$-directed iff every collection of fewer than $\kappa$-many elements in $P$ has an upper bound in $P$. $P$ has the $\mathbf{\kappa}$ chain condition iff ...

**8**

votes

**1**answer

328 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 ...

**0**

votes

**1**answer

294 views

### Terminology for posets.

What should I call a poset with the property that each element has AT MOST ONE
predecessor?
(I'm actually interested in the special case in which there are no infinite descending chains.)
...

**1**

vote

**2**answers

945 views

### On the number of antichains of a poset

I am not an expert in combinatorics, but I need to count (or to approximate) the number of antichains of a poset.
The idea relies on approximating this number by embedding my poset into another one, ...

**9**

votes

**1**answer

309 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 ...

**5**

votes

**1**answer

363 views

### Does “antichain” mean something different in set-forcing than in lattice theory?

On page 3 of Introduction to Lattices and Order, Davey and Priestley define an antichain in a poset $\langle P,\leq\rangle$ as a set of pairwise incomparable elements:
The ordered set P is an ...

**7**

votes

**3**answers

587 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) ...

**2**

votes

**2**answers

506 views

### A problem about posets similar to Suslin's problem

Suslin's problem is:
Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$?
The answer is that it's independent of ZFC. The related ...

**3**

votes

**2**answers

644 views

### Does order isomorphism of linear extensions of two partially ordered sets imply order isomorphism of themselves?

Consider two partially ordered sets $A = \{a< b,a< c\}$, $B=\{x< z,y< z\}$.
Their linear extensions (here we allow equality in linear extensions) for $A, B$ are
$$A_L=\{A_1=\{a< b< ...

**2**

votes

**1**answer

278 views

### Count of lattices on finite set

Let $p(n)$ denote count of lattices on finite set $G$, $|G|=n$ (without isomorphism). It's know closed formula for $p(n)$?
It's clear, that $1 \leq p(n)$ and also that $p(n-1) \leq p(n)$ for $n \geq ...

**4**

votes

**1**answer

295 views

### Posets of finite sequences are highly connected

I need the following result for an example in a paper I'm writing. It's easy enough to prove, but I'd prefer to just give a reference. Does anyone know one?
Fix $1 \leq k \leq n$. Define $X_{n,k}$ ...

**3**

votes

**0**answers

358 views

### 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 ...

**14**

votes

**3**answers

3k 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 ...

**7**

votes

**2**answers

385 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 ...

**14**

votes

**3**answers

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?

**2**

votes

**1**answer

257 views

### Selecting k sub-posets

I ran into the following algorithmic problem while experimenting with classification algorithms. Elements are classified into a polyhierarchy, what I understand to be a poset with a single root ...

**51**

votes

**6**answers

11k 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 ...

**14**

votes

**3**answers

822 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 ${\leq}$, ...

**2**

votes

**0**answers

122 views

### non-degenarete tools to calculate a derived functor on a model category which is a poset?

Are there theorems (esp. computational tools) on model categories which survive and do not trivialise when its underlying category is a (quasi-)poset ? Are there tools
that may help to calculate ...

**1**

vote

**2**answers

898 views

### Proof of glb and lub of Lexicographic Product of poset

Is there a book, or a paper where the Lexicographic glb and lub are proven commutative, associative, idempotent and absorbing. I have already proven this, but would like to check proofs, and a short ...

**4**

votes

**4**answers

768 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

**3**answers

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 ...

**7**

votes

**1**answer

509 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 ...

**4**

votes

**2**answers

427 views

### Is every lattice the fixed-point set of an order endomorphism of ⋄^n?

(Asked by Nathaniel Hellerstein on the Q&A board at JMM)
Let ⋄ be the 4 element lattice
τ
/ \
i j
\ /
f
Is every lattice isomorphic to the fixed point lattice of some ...