The order-theory tag has no wiki summary.

**7**

votes

**1**answer

316 views

### What should the morphisms in the Category of Directed Sets be?

Directed sets are defined to be sets equipped with a preorder that admit (finitary) upper bounds e.g. pairs $(D, \preceq)$ such that $\forall p,q \in D$ there exists $r \in D$ such that $p \preceq r$ ...

**4**

votes

**2**answers

409 views

### Projections in a W*-algebra as a continuous lattice?

A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed ...

**2**

votes

**1**answer

108 views

### References on countable W*-algebras

In "Operator algebras with a faithful weakly-closed representation" (1955), Kadison describes a countable W*-algebra as a C*-algebra which has a faithful representation as a countably decomposable ...

**16**

votes

**2**answers

915 views

### Can all $\aleph_2$-dense subsets of $\mathbb{R}$ be isomorphic?

Let $\kappa$ be an infinite cardinal. For a subset $A \subseteq \mathbb{R}$, we say that $A$ is $\kappa$-dense if $|A \cap (a, b)| = \kappa$ for every interval $(a, b)$. By Cantor, any two ...

**21**

votes

**3**answers

552 views

### Is it possible to construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself?

Is it possible to construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself?

**11**

votes

**2**answers

444 views

### How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?

How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
How many subsets of the long line $\omega_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$?
I can see that results in ...

**12**

votes

**2**answers

340 views

### What is the least ordinal than cannot be embedded in $\mathbb{R}^\mathbb{R}$?

Let $\mathbb{R}^\mathbb{R}$ be the set of functions $\mathbb{R}\to\mathbb{R}$ patially ordered by eventual domination. Obviously, every ordinal below $\omega_1$ can be embedded in ...

**10**

votes

**2**answers

350 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 ...

**23**

votes

**2**answers

460 views

### Order type of the smallest set containing the identity function and closed under exponentiation

Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapsto ...

**16**

votes

**2**answers

275 views

### Is it possible to reconstruct an order type from its initial segments?

Suppose
$T$ is a totally ordered set without a maximal element,
$\tau$ is the order type of $T$,
$S$ is the set of order types of all proper initial segments (downward closed subsets) of $T$.
Is ...

**10**

votes

**1**answer

230 views

### Order dimension and weak poset partitions

The order dimension of a poset $(P,\leq)$ is the least number of linear extensions of $(P,\leq)$ such that the intersection of these extensions is $(P,\leq)$. The wikipedia entry provides some ...

**1**

vote

**0**answers

94 views

### Set of upper bounds is finite for any finite subset

Is there a term to describe a preordered set $P$ in which any finite subset $S \subset P$ has at most finitely many minimal upper bounds? The preordered sets I'm studying generally aren't ...

**0**

votes

**2**answers

279 views

### Order-isomorphic down-set lattices

Let $X$ be an ordered set. A down-set (also called a lower set or an order ideal) of $X$ is a subset $D$ of $X$ such that for every $x, y \in D$, if $x \in D$ and $y \leq_X x$, then $y \in D$. The ...

**0**

votes

**1**answer

125 views

### Counting linear extensions of unlabeled series parallel structures

I am interested in the problem of counting the number of linear extensions of series-parallel structures. The wikipedia article at http://en.wikipedia.org/wiki/Series-parallel_partial_order pointed me ...

**5**

votes

**1**answer

257 views

### Rotation-invariant strict-inclusion-preserving preorderings on subsets of the circle

Say that a preordering $\le$ on a set of subsets of some space preserves strict inclusion provided that $A\lt B$ whenever $A\subset B$ (where $A\lt B$ iff $A\le B$ and $B \not\le A$).
Let the space ...

**4**

votes

**4**answers

472 views

### Why do we choose the standard total order on the integers?

I understand why the set of natural numbers $\mathbb N = \{ 0, 1, 2, \cdots \}$ is equipped with a total order. Indeed, every monoid has a pre-order, where $$n' \succeq n \quad \mathrm{if~and~only~if} ...

**0**

votes

**2**answers

246 views

### Reference Book for supremum and infimum theorems

For my work I need many of the very easy and basic properties of suprema and infima. While they are all pretty easy to prove, I would prefer to refer to a standard text book. However I did not find ...

**1**

vote

**0**answers

60 views

### Looking for a uniform explanation of algebras with canonical generators.

Let $\mathcal{V}$ be a finitary variety i.e. the algebras for a signature whose operations have finite arity and for some arbitrary set of equations. Then any algebra $A \in \mathcal{V}$ has a ...

**2**

votes

**1**answer

136 views

### Distributive lattice embedding into a finite lattice.

Suppose one has an inclusion $\iota : D \hookrightarrow S$ where $D$ is a finite distributive lattice and $S$ is a finite join-semilattice.
If $\iota$ preserves all meets and joins one can show that ...

**12**

votes

**2**answers

428 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**

votes

**2**answers

337 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 ...

**0**

votes

**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.

**1**

vote

**1**answer

132 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 ...

**4**

votes

**2**answers

315 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 ...

**1**

vote

**0**answers

198 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 ...

**2**

votes

**1**answer

172 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

268 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 ...

**2**

votes

**1**answer

237 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**

votes

**1**answer

252 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 ...

**4**

votes

**1**answer

366 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< ...

**4**

votes

**2**answers

359 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 ...

**3**

votes

**2**answers

166 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 ...

**0**

votes

**0**answers

227 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 ...

**0**

votes

**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 ...

**1**

vote

**0**answers

127 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 ...

**2**

votes

**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, ...

**2**

votes

**3**answers

307 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 ...

**1**

vote

**1**answer

208 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 < ...

**5**

votes

**1**answer

333 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

424 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

465 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

389 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

268 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

427 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

642 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

345 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 ...

**6**

votes

**1**answer

539 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 ...