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**5**

votes

**1**answer

336 views

### Characterizing $\omega_1$-like dense linear orderings

I recently came upon the following theorem which was attributed to J. Conway:
For each $A\subset \omega_1$, let $\Phi(A)$ be a linear ordering of type $\sum_{\alpha<\omega_1} \tau_\alpha$, where ...

**6**

votes

**1**answer

430 views

### Strictly order preserving maps into the integers

If $P$ and $P'$ are partial orders, a strictly order preserving map from $P$ to $P'$ is an $f:P\to P'$ satisfying that $x\lt y$ implies $f(x)\lt f(y)$ for all $x,y\in P$.
An interval in $P$ is a set ...

**3**

votes

**2**answers

468 views

### Cantor theorem on orders

It is "a well-known theorem of Cantor", said Sierpinski (circa 1920), that every countable total order can be imbedded in the rationals, and he proceeds to demonstrate that, assuming the continuum ...

**0**

votes

**1**answer

391 views

### A property of a product of posets

Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a
\curlyvee b$ if only if there is a non-least element $c$ such that $c
\leqslant a \wedge c \leqslant b$.
I call a poset ...

**1**

vote

**1**answer

215 views

### Order density of smooth functions among continuous functions?

Let $\mathcal{C}^0([a,b],\mathbb{R})$ be the space of all continuous functions $f:[a,b]\rightarrow\mathbb{R}$ and $\mathcal{C}^\infty([a,b],\mathbb{R})$ the subspace of all smooth functions. Define ...

**2**

votes

**1**answer

269 views

### Is there a Dirichlet Unitary Unit Theorem?

Dirichlet's unit theorem computes the group of units of the algebraic numbers of a number field. There are a few generalisations for orders available.
Assume the order has an involution. For example, ...

**6**

votes

**0**answers

430 views

### 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 ...

**12**

votes

**1**answer

643 views

### Does every feasible partial order relation on the natural numbers extend to a feasible linear order relation?

It is well known that every partial order on a set can be extended
to a linear order on that set. That is, for every partial order
$\lhd$ on a set $X$, there is a linear order $\prec$ on $X$ such
that ...

**7**

votes

**3**answers

322 views

### Extracting countable chains from linear orders

There is a well-known fact in infinite combinatorics asserting that for each infinite linear order $P$ there is a countable subset $R\subseteq P$ of order type either $\omega$ or $\omega^{*}$
(by ...

**1**

vote

**1**answer

349 views

### 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 ...

**4**

votes

**0**answers

194 views

### Oriented matroids and posets?

Is there a characterization of oriented matroids in terms of order theory, similar to that of matroids as geometric lattices?
Does this question make sense at all? I have seen (for instance in ...

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votes

**1**answer

541 views

### A categorical characterization of the lexicographic order

In $Pos$ (the category of partial ordered sets and order preserving maps) there is the categorical product of two objects, but on the set product there is (naturally) also the lexicographic order. I ...

**0**

votes

**3**answers

956 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 $P$, ...

**7**

votes

**1**answer

605 views

### 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 ...

**3**

votes

**5**answers

638 views

### Visualizing large posets

Hi, does somebody know if there is any software for visualizing very large posets? (like those in page 27 of this notes of Guenter Ziegler). They may arise (as in that text) by considering the face ...

**4**

votes

**2**answers

303 views

### Heights of several interesting posets

Let the height of a poset $P$ be the supremum of ordinals that are order types of all well-ordered subsets of $P$ (with order inherited from $P$).
Define several sets of total functions, in each ...

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votes

**2**answers

336 views

### Ordinals and complexity classes

What is the least recursive ordinal $\alpha$ such that there is no algorithm in complexity class $\mathsf{P}$ which implements a well-ordering of $\mathbb{N}$ with order type $\alpha$? (where the size ...

**3**

votes

**0**answers

210 views

### When Aut(M) preserves a linear order?

I have a general-type question:
Suppose $M$ is a countable structure that is ultrahomogeneous, i.e. every (partial) isomorphism between finitely generated substructures of $M$ extends to an ...

**9**

votes

**0**answers

198 views

### 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 ...

**1**

vote

**2**answers

402 views

### Definition of $\beta$-limit ordinals

Hello,
I am reading Rosenstein's "Linear Orderings" and I am not sure if I am missing something, or if there is an error.
He gives the definition of a $\beta$-limit ordinal inductively, as follows ...

**1**

vote

**2**answers

257 views

### Automorphisms of locally finite countable posets-2

Given is a locally finite countable connected poset which satisfies further the following properties:
Let $C$ be any maximal chain ( i.e. inextendible chain) and $A$ be any antichain. Then $A$ is ...

**1**

vote

**1**answer

369 views

### Causal sets quantization

Hi,
Here are a couple of questions:
Is there a way to classify all homomorphisms between two finite posets?
Same question as (1) but for infinite, locally finite countable connected posets.
...

**4**

votes

**1**answer

247 views

### Automorphisms of locally finite countable posets

I rephrase my last question. Given a locally finite countable connected (as a graph) poset which satisfies the following further condition: the intersection of any antichain with the set of elements ...

**2**

votes

**1**answer

214 views

### Automorphisms of locally finite countable posets

Hi,
Is the automorphism group of a countable locally finite connected poset finite or countable?
If not, is there a way to equipp it (the uncountable group) with a topology and a measure?
Need this ...

**1**

vote

**0**answers

206 views

### 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 ...

**2**

votes

**2**answers

261 views

### 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 ...

**4**

votes

**1**answer

422 views

### Intervals in posets: how to extend interval orders, Allen's algebra, and interval graphs to intervals of posets?

BACKGROUND
Assume a poset $\langle P, \le \rangle$. For two points $a,b \in P$
with $a \le b$, then $I = [a,b] = \{ x : a \le x \le b \}$ is the
interval between $a$ and $b$.
When $P$ is a chain ...

**10**

votes

**1**answer

789 views

### Discrete version of Nullstellensatz?

Hi. I was reading the paper "On the foundations of combinatorial theory (VI): The idea of a generating function" by Doubilet, Rota and Stanley, and there is a relation treated which is very ...

**0**

votes

**1**answer

307 views

### Power of an order relation

Let there be > included in AxB as a binary relation.
What does (x)>^2(y) mean? What is the meaning of an order relation raised to a power?
My first tought was that >^2 = >x> which is a cartesian ...

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votes

**1**answer

264 views

### is $ded^{*}(\kappa)< ded(\kappa)$ consistent?

Hello,
I wonder if anyone knows this.
Definition:
$ded\left(\lambda\right)$ is the supremum of all sizes
of linear orders with a dense subset of size $\lambda$.
$ded^{*}\left(\lambda\right)$ is ...

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votes

**2**answers

698 views

### Is $Ded(\kappa)<Ded(\kappa)^\omega$ consistent?

Hello,
I want to ask if anyone can tells us what is known (consistently) about $Ded(\kappa)$, $\kappa$ an infinite cardinal.
Definition If there is a dense linear order w/o endpoints of size ...

**3**

votes

**1**answer

561 views

### Height of ordered set

In all "constructive" fixed-point theorems for functions on ordered sets that I am aware of, where the fixed point is obtained as the limit of a stationary increasing transfinite sequence, it is ...

**6**

votes

**1**answer

1k views

### Probability distributions: The maximum of a pair of iid draws, where the minimum is an order statistic of other minimums?

General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n ...

**0**

votes

**2**answers

859 views

### order-preserving question

A B are totally-ordered sets and there exist two maps f and g such that f is a order-preserving injection from A to B and g is a order-preserving injection from B to A.
Q: Are A and B necessarily ...

**8**

votes

**1**answer

591 views

### Which countable linear orders are $\aleph_0$-categorical?

The question is: Which countable linear orders are $\aleph_0$-categorical?
I have a bit of progress on this:
Define a discrete tuple to be a set of elements, ordered discretely, such that if $a$ and ...

**12**

votes

**1**answer

643 views

### Converse to Banach’s fixed point theorem for ordered fields?

Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := ...

**7**

votes

**1**answer

2k views

### Number of graphs with a given number of nodes, edges and triangles

Hi. Does anyone know if it is possible to enumerate the set of labeled/unlabeled graphs (loopless, undirected, only one edge between pairs of nodes) having a given number of nodes, edges and ...

**0**

votes

**3**answers

462 views

### way-below relation

Wiki says that an infinite set may be way-below another set.
More precisely, in a power set, pow(S), with subset order and with X and Y subsets of S, does X way-below Y entail that X is finite?
...

**5**

votes

**1**answer

518 views

### Bridge game with only one suit: strategy

This game looks like bridge, but 1- there are only two players Alice and Bob, 2- there is only one suit, whose cards are numbered $1, 2,\ldots,2n$. One deals each player $n$ cards. Therefore Alice ...

**0**

votes

**2**answers

911 views

### Topological sort of partial order into sorted sets

Given a partial order of elements, one can use topological sorting to produce a sorted list of elements. For example, if we have the partial order A->B and A->C, then the possible topological sort ...

**3**

votes

**2**answers

583 views

### Is it reasonable to define `poset homotopy' as a `natural transformation of posets'?

Pausing to speak metaphorically for a bit, I have always thought of partial orders as being something like a simplified version of topological spaces. (In the finite case at least, all topological ...

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votes

**4**answers

851 views

### Universal order type

Every countable order type, such as the countable ordinals, $\mathbb Z$, etc. can be embedded in $\mathbb Q$, so it is universal for countable order types. Is there a universal space for all linear ...

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votes

**0**answers

78 views

### A specific notion between the notions of transversal and system of distinct representatives.

Let $X$ be a set, let $\mathcal{C}$ be a collection of subsets of $X$, and let $x_1, \dots , x_k \in X$. Say that the sequence $\{x_i\}_{i=1\dots k}$ is a sequential transversal (of length $k$) ...

**0**

votes

**2**answers

356 views

### Semilattices with n elements

How many n-element semilattices there are?
For example, for n-element partially ordered set we can figured out, that there are $2^{n*(n-1)}$ possible sets.
And can I find all possible n-element ...

**3**

votes

**1**answer

863 views

### Well ordering of countably branching well founded trees

Hello everybody,
I would like to say, before stating the question, that I'm not a mathematician, and therefore i apologize in advance if the question is trivial, or well-known. I'll try to state it ...

**8**

votes

**1**answer

389 views

### Does this “flipping lexicographic” ordering have a standard name?

I’ve run into the following straightforward variant of lexicographic ordering, and am wondering if it has a standard name. I’ve been calling it the flipping lexicographic ordering, for evident ...

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vote

**0**answers

223 views

### Modern books about orders and algebras on trees

Please help to find books about orders and algebras on trees.
If there is no modern books, please advice good old ones!
I'm more interested in finite trees (my current problem), but infinite ones are ...

**8**

votes

**1**answer

333 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 ...

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votes

**2**answers

192 views

### Of what kind of complemented bounded poset are the structures in my quasi-variety?

I feel that my question is very basic, but, somewhat suprisingly, nobody was able to give me an answer so far:
Let
$\mathbf{M} := \langle \{ 0,1 \}, 0, 1, \leq, \neg \rangle$
be the structure with ...

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votes

**2**answers

618 views

### Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?

It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can ...