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337 views

### Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?

Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
Here is one natural ...

**2**

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**2**answers

328 views

### Banach lattice subspace of $C([0,1])$ not a sublattice

This is probably easy, but I did not see it in standard texts. Describe a closed subspace $V$ of $C([0,1])$ such that $V$ is a Banach lattice (in the pointwise ordering), but $V$ is not a sublattice ...

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**1**answer

290 views

### Lattice ordered group

Does there exist a lattice-ordered group of rational rank $1$?
This is true for totally ordered group.

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**1**answer

128 views

### Covering of a partial order by upwards convex sets

First off: I'm not an expert in order theory, so some of my terms might be off; correct them if you wish.
Let me call a subset $A$ of a lattice $(S,\le)$ upwards convex (not sure if that's actually ...

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**2**answers

299 views

### Cardinality of Equivalence Relation of Eventually Sublinear Functions

Let $\Bbb{R}^{+}\_{0}$ be the set of non-negative real numbers and $\Bbb{R}^{+}$be the set of positive reals. Let us say that a function $f \colon \Bbb{R}^{+}\_{0} \to \Bbb{R}^{+}\_{0}$ is eventually ...

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**0**answers

193 views

### existence of order preserving map [closed]

suppose A is a linear order set with a copy of rationals in it that is $A=B\cup\{\bar{r}:r\in\mathbb{Q}\cap[0,1]\}$. is there an orde preserving map that preservs sup and inf betwwen A and $[0,1]$ in ...

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votes

**1**answer

170 views

### Is there research on the notion of co-accessibility?

I want to start off with a disclaimer that I am only a mathematical amateur. Please forgive me for ignorance or any non-standard nomenclature I use here :)
Let's start off with some context.
Let X ...

**5**

votes

**1**answer

244 views

### Complete anti-chain lattices and the axiom of choice

Hello, everyone. I'm trying to find out about lattices of anti-chains, and was wondering whether you could help me with getting to grips with a Comp. Sci. paper I'm struggling with.
I've been reading ...

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**1**answer

228 views

### Cardinality of group of order-preserving functions from R to R

The title pretty much says it all. What is the cardinality of $G$, the group of all functions $f: \mathbb{R} \to \mathbb{R}$ such that $\forall x,y\in \mathbb{R} \left( x>y\Rightarrow ...

**6**

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**1**answer

246 views

### Is it possible to decide in polynomial time if a poset is a subposet of another which is given ?

I am reading some theory on partial orders and I wonder something which perhaps has a simple answer : Given two partial orders $G_1,G_2$ (by their hasse diagrams), is it possible to know in ...

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votes

**1**answer

345 views

### Uncountable orderings

Let $P$ be an uncountable linear ordering. Is it true that either $P$ contains an order-copy of $\omega_1$ or there is $x_0\in P$ such that there exist uncountably many distinct $y\in P$ with $y< ...

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330 views

### Definition of continuous functions in order theory

If we have a complete partial order (i.e. directed complete) I find frequently the following definition of a continuous function. A function $f:A\to B$ where $A$ and $B$ are cpos is called continuous ...

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**2**answers

162 views

### A linearly orderable monoid which does not embed into a linearly orderable group

It is known (after an example of A.I. Mal'cev) that there exist cancellative semigroups which do not embed into a group. On the other hand, it is not difficult to see that every linearly orderable ...

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**0**answers

223 views

### Transitive closures and inductive reasoning [solved]

Let's say that r is an endorelation over A (i.e. $r$ is a subset of $A \times A$), $\bar{r}$ is the transitive closure of r (i.e. the least set containing r and being transitive).
Furthermore $r$ has ...

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**1**answer

170 views

### wellfounded sets and predecessors

Following question: Let's assume that W is a wellfounded set, i.e. it has a partial order and every nonempty subset of W has minimal elements with respect to the order.
Now we can easily define a ...

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**0**answers

125 views

### How many more join-irreducibles can there be in a sub join-semilattice of a finite lattice?

Let $L$ be a finite lattice. Then $L$ is generated by its join-irreducible elements $J(L)$ or alternatively its meet-irreducible elements $M(L)$.
If $S \subseteq L$ is a sub join-semilattice then ...

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**0**answers

117 views

### Generalization of order dimension and interval orders

A partial order $(X, <_X)$ has order dimension $n$ if it can be realized as
the product order of $n$ total orders, which means that there is an
order-embedding between $(X, \lt_X)$ and $(Y^n, ...

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**3**answers

305 views

### A characterisation of Boolean algebras

Let $M$ be a meet-semilattice with a least element $0$. Suppose there is an order-reversing involution $a \mapsto -a$ on $M$ such that for all $a, b \in M$, $a \wedge b = 0$ if and only if $b \le ...

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**1**answer

199 views

### Strictly totally ordered semigroups - Looking for references

Let $\mathfrak A = (A, \cdot)$ be a semigroup (written multiplicatively). We say that $\mathfrak A$ is linearly orderable if there exists a total order $\le$ on $A$ such that $ac < bc$ and $ca < ...

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**1**answer

323 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 ...

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**1**answer

412 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 ...

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456 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 ...

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382 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 ...

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**1**answer

206 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 ...

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**1**answer

266 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, ...

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412 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 ...

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**1**answer

628 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 ...

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**3**answers

319 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 ...

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**1**answer

332 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 ...

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**0**answers

190 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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**1**answer

530 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 ...

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895 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 $P$, ...

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601 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 ...

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**5**answers

592 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 ...

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**2**answers

298 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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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 ...

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209 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 ...

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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 ...

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398 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 ...

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252 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 ...

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**1**answer

366 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.
...

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**1**answer

244 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 ...

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**1**answer

209 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 ...

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204 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 ...

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252 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 ...

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408 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 ...

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777 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 ...

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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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**1**answer

261 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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690 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 ...