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### Which linearly ordered sets have the property that their completion is equipotent with their powerset?

As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the ...

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**1**answer

82 views

### Monotonic sequence (edited) [closed]

For any two n-dim vector $v$ and $v'$ define $v\leq v'$ iff for each $1\leq i\leq n$, $v_i\leq v_i'$.
Suppose further that the entry of vectors can only take values from $m$ distinct values $\{a_1, ...

**3**

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**1**answer

87 views

### existence of more distributive Boolean lattices

Are there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that
$$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in ...

**1**

vote

**1**answer

420 views

### Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based
on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...

**2**

votes

**1**answer

137 views

### Characterizing Inf and Sup sets

For a poset $(X,R)$, where $R$ is a partial order on $X$,
let $\operatorname{Inf}(R)$ be the set of all $A\subseteq X$ which have an infimum in $(X,R)$.
let $\operatorname{Sup}(R)$ be the set of all ...

**4**

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**0**answers

68 views

### Reference for statement that almost every $n$-element partial order has trivial automorphism group

I'm looking for a reference for the statement that almost every partial order on $n$ elements has trivial automorphism group. I've been told that this is a folklore result. Does anyone know of a ...

**1**

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**0**answers

64 views

### First-order Peano Axioms and order-completeness of $\mathbb{N}$ [closed]

Definition: An ordered set is order-complete if any nonempty subset with an upper bound, has a lowest upper bound or supremo.
Notation: We denote the system of first-order Peano Axioms (along with ...

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votes

**1**answer

167 views

### Semitransitive relations

By a digraph, let us mean an ordered pair $(X,r)$ with $r : X \times X \rightarrow B,$ where $X$ is a set and $B = \{\mathrm{False}, \mathrm{True}\}.$
Then supposing $\mathbb{X} =(X,r)$ is a digraph, ...

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**0**answers

87 views

### A categorical analogue of Debreu's independent factors theorem

Background
A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...

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votes

**1**answer

101 views

### Has the single sorted case of formal concept analysis been investigated?

A formal context in formal concept analysis is a triple $K = (G, M, I)$ where $G$ is a set of objects, $M$ is a set of attributes and the binary relation $I \subset G \times M$ shows which objects ...

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122 views

### name for a subset of a binary relational structure which is “closed downward”?

Let $X$ be a set and suppose that $R$ is a binary relation on $X$. Suppose further that $S\subseteq X$ has the property that whenever $y\in S$ and $xRy$ ($x\in X$), then also $x\in S$. Does such an ...

**1**

vote

**1**answer

119 views

### Uniqueness of minimal completions of a partially ordered set

The partially ordered set $(Y,\le)$ is called a superspace of the partially ordered set $(X,\le')$ iff
$X\subseteq Y$.
$\le'=(\le\cap X^2)$.
and a completion of $(X,\le')$ if in addition
$~~ 3$. ...

**2**

votes

**1**answer

93 views

### Example involving partially ordered Abelian groups

Definition 1:
Let $(G,\leq)$ be a nonzero partially ordered Abelian group with order unit $u$. (Recall that $u\in G$ is a order unit if, for every $g\in G$, there exists $N\in\mathbb N$ such that ...

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votes

**1**answer

260 views

### Is any order bounded continuous linear functionals a difference of positive continuous functionals?

Let $B$ be a Banach space and $K$ a closed proper cone in $B$ such that the induced partial order makes $B$ a vector lattice. Let $K'=\{x^*\in B':\langle x^*, x\rangle\geq 0\ \forall x\in K\}$ the ...

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755 views

### An order type $\tau$ equal to its power $\tau^n, n>2$

(This is a re-post of my old unanswered question from Math.SE)
For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types.
Recall that:
...

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votes

**1**answer

131 views

### Totally right preorderable groups

Are there any known non-trivial sufficient conditions, or full characterizations, of a totally right-preorderable group?
More precisely:
totally right-preorderable: has a non-trivial total ...

**3**

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**0**answers

92 views

### Expected size of $k$-th layer of a POSET

Is this known?
What is the expected width of the $k$-th layer (anti-chain layer) of a $d$-dimensional partially ordered set of $n$ elements formed by product of $d$ random linear orders chosen from ...

**5**

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**0**answers

187 views

### Proving equivalence of a tree-based version of Countable Choice for families of finite sets

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5:
Each of the following statements imply those beneath it.
The countable union of ...

**2**

votes

**1**answer

132 views

### Counting a type of self-dual poset

An exercise in Stanley's Enumerative Combinatorics (Chapter 3, ex. 8) asked for an example of a finite self-dual poset, (i.e. there is a bijection $f: P\to P$ such that $s\le t \Longleftrightarrow ...

**4**

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**1**answer

186 views

### Does every locally finite acyclic directed set embed into a linear order locally isomorphic to the integers? (Edit: extend, not merely embed.)

Let $S=(S,\prec)$ be a set together with an acyclic binary relation, generally nontransitive. $S$ is locally finite if, for every element $x\in S$, the sets $\{w|w\prec x\}$ ("direct past of $x$") ...

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**1**answer

240 views

### Does every countably infinite interval-finite partial order embed into the integers?

A partially ordered set $(S,\le)$ is called interval finite if the open intervals $(x,z):=\{y|x\le y\le z\}$ are finite for all choices of $x,z$ in $S$. An embedding $(S,\le)\rightarrow(S',\le')$ of ...

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**1**answer

144 views

### Linear order extensions on (nonabelian) groups

If $G$ is a group with a (left) linear order, does every (left) partial order on $G$ extend to a (left) linear order?
The answer is affirmative on abelian groups, where being torsion-free is ...

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**2**answers

224 views

### Conditions for a group to be lattice-ordered

Given a set $S$ with a group operation $\cdot$ and a lattice ordering $\leq$, I wish to know when we can say that $\cdot$ preserves $\leq$, i.e. $(x\vee y)z=xz\vee yz$ and similarly for meets.
...

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329 views

### Extending a partial order while preserving an automorphism

It is well known that if $(P, \leq)$ is a partial order then $\leq$ can always be extended to a linear order. This is sometimes called Szpilrajn´s theorem although it had been previously proved by ...

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369 views

### Embedding a brouwerian lattice into a boolean lattice

I have already asked a similar question at
http://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra
but have received no answer.
Sorry, I ask a ...

**3**

votes

**2**answers

202 views

### When is a filter generated by a (countable) chain?

In any partial order $(P,\leq)$ it is easy to see that every chain generates (i.e., by taking the upwards closure) a filter, and any filter built as a result of the Rasiowa-Sikorski lemma in forcing ...

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387 views

### A characterization of the poset of filters on a set

For the lattices of all subsets of a given set it is known an axiomatic characterization: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra.
The ...

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181 views

### Maximal chains in a quasi-order of linear order types

Let $\mathcal{T}_\kappa$ be the set of all linear order types of cardinality $\kappa$. Let $\prec$ denote a binary relation on $\mathcal{T}_\kappa$ representing embeddability of order types (note that ...

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275 views

### What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...

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134 views

### For what classes of comparability graphs are their complements also comparability graphs?

An interval graph is an intersection graph of real intervals, that is, an undirected graph whose vertices can be labeled with real intervals so that there is an edge between two vertices iff their ...

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209 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 ...

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312 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 $p \preceq r$ ...

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385 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 ...

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106 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 ...

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874 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 ...

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537 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?

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440 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 ...

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**2**answers

332 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 ...

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342 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 ...

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445 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 ...

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268 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**

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**1**answer

226 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 ...

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91 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 ...

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262 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 ...

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**1**answer

122 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 ...

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255 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 ...

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459 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} ...

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237 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 ...

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### 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 ...

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**1**answer

130 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 ...