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6
votes
1answer
250 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
4answers
410 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
2answers
198 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
0answers
45 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
1answer
112 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 ...
9
votes
1answer
289 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
2answers
281 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
1answer
280 views

Lattice ordered group

Does there exist a lattice-ordered group of rational rank $1$? This is true for totally ordered group.
1
vote
1answer
111 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
2answers
219 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
0answers
162 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
1answer
165 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
1answer
196 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
1answer
211 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
1answer
218 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
1answer
286 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< ...
3
votes
2answers
244 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
2answers
151 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
0answers
206 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
1answer
158 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
0answers
112 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 ...
1
vote
0answers
104 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
3answers
289 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
1answer
180 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 < ...
4
votes
1answer
286 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
1answer
364 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
2answers
405 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
1answer
356 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
1answer
188 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
1answer
246 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
0answers
350 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
1answer
533 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 ...
-2
votes
1answer
637 views

A question regarding the Continuum Hypothesis

Consider the following theorem of Cantor: If one has a simply-ordered set (M.<) which fulfills the three conditions: a) |M| = aleph-null; b) M has no greatest or least element; c) M is everywhere ...
7
votes
3answers
303 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
1answer
280 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 ...
3
votes
0answers
180 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
1answer
489 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
3answers
664 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 $P$, ...
6
votes
1answer
596 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
5answers
484 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
2answers
288 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 ...
7
votes
2answers
330 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
0answers
205 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
0answers
194 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
2answers
349 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 ...
0
votes
2answers
244 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
1answer
356 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
1answer
231 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
1answer
198 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
0answers
191 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 ...