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4
votes
0answers
223 views

Order theory as a foundation of mathematics?

I know the followings kinds of formalization of mathematics: based on set theory (e.g. ZFC) based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example) based on category ...
1
vote
0answers
39 views

Is the order convergence topology on a poset always Hausdorff?

In this post two topologies on a poset $(P,\leq)$ were defined: the interval topology $\tau_i(P)$ and the order convergence topology $\tau_o(P)$. It turns out that $\tau_i(P)$ is always $T_1$ and ...
2
votes
1answer
56 views

Interval topology and order convergence topology

Throughout this post, let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where ...
4
votes
2answers
154 views

Counterexample on completely distributive lattices

I would like to see an example of a complete lattice $C$ which is both a frame and a dual-frame, i.e. finite meets distribute over arbitrary joins and finite joins distribute over arbitrary meets ...
-2
votes
2answers
59 views

does a lattice have a minimal item [closed]

A lattice (L, ≤) is a partially ordered set, where each two elements of the set have a least upper bound and a greatest lower bound. Let's consider the situation where L is finite. I think that ...
0
votes
0answers
40 views

Quasi-transitive decomposition of a transitive graph

Let $G=(V,E)$ be a simple digraph that is semi-complete (ie. there's at least one arc between each unordered pair of vertices) and quasi-transitive (ie. its complement is transitive). Is it true that ...
1
vote
1answer
39 views

monotonicity alike functions

assuming we have two smooth function ${f_1},{f_2}:{R^N} \to R$, under what condition, we have ${f_1}\left( {{{\bf{x}}_1}} \right) \ge {f_1}\left( {{{\bf{x}}_2}} \right) \leftrightarrow ...
2
votes
0answers
26 views

Conditions for monotone function to take maximal chains to maximal chains surjectively

Suppose that $P$ and $Q$ are graded posets (with rank function $r$) and suppose that all maximal chains of $P$ and $Q$ have length $n$. Let $f:P \to Q$ be a surjective monotone function such that ...
1
vote
0answers
54 views

Is $\mathcal{P}(\omega)/fin$ with the interval topology path-connected?

Is $\mathcal{P}(\omega)/fin$ with the interval topology path-connected? (You find the definition of $\mathcal{P}(\omega)/fin$ here.)
1
vote
2answers
98 views

Continuous image relation on topological spaces

Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by for $X, Y \in ...
2
votes
1answer
166 views

Normal subgroup of a totally ordered group

A totally ordered group is a group equipped with a compatible total order, that is, $x\leq y$ and $z\leq t$ imply $x+z\leq y+t$ for all $x,y,z,t$ in the group. Is it true that every totally ordered ...
5
votes
1answer
106 views

Is an open map with open relative diagonal necessarily a local homeomorphism?

Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta_f : X \to X \times_Y X$ is an open embedding of locales. Does it follow that $f : X \to Y$ is a ...
3
votes
1answer
160 views

Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space?

Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space? (You find the definition of $\mathcal{P}(\omega)/fin$ here.) Remark: According to this, the interval topology of ...
2
votes
3answers
151 views

Antichain on $\mathcal{P}(\omega)/fin$ of cardinality $2^{\aleph_0}$?

We define an equivalence relation on $\mathcal{P}(\omega)$: for $x,y\in\mathcal{P}(\omega)$ we say $$x\simeq_{fin} y \text{ iff there is } n \in \omega \text{ such that } x\setminus \{0,\ldots,n\} = y ...
4
votes
1answer
69 views

Is $\{0,1\}^\omega$ the order-preserving image of $\{0,1\}^\omega$ modulo some finiteness relation?

Consider the following equivalence relation on $\{0,1\}^\omega$: $x\simeq y$ iff there is $n\in\omega$ such that $x(k)=y(k)$ for all $k\in\omega$ with $k\geq n$. It is easy to see that ...
4
votes
1answer
181 views

Characterising subsets of the reals as ordered spaces

There are concise and elegant characterisations of the real line as a topological space and as an ordered space in the literature. I am interested in the harder case of characterising subsets of the ...
2
votes
2answers
98 views

Completion of a single totally ordered down-set

This is a follow-up question to Complete sets of incompatible totally ordered down-set in a partially ordered set. Let $(P,\leq)$ be a partially ordered set such that for every $p\in P$ the set ...
3
votes
2answers
165 views

Complete sets of incompatible totally ordered down-set in a partially ordered set

Let $(P,\leq)$ be a partially ordered set. A down-set is a set $d\subseteq P$ such that $x\in d$ and $x'\in P, x'\leq x$ imply $x'\in d$. If the down-set is totally ordered, we say it is a totally ...
3
votes
1answer
117 views

Directed Hypercube Minimal Cuts

If $[n]:=\{1,2,\ldots, n\}$ for some $n\in\mathbb{N}$, then the hypercube digraph of dimension $n$, denoted $H_n$, is the graph whose set of vertices is the power-set $\wp([n])$ where two vertices ...
1
vote
2answers
80 views

Order-preserving image of a complete lattice

If $L$ is a complete lattice and $P$ is a poset and $f: L\to P$ is an order preserving surjective map, does this imply that $P$ is a (complete) lattice?
2
votes
1answer
75 views

Order-preserving images of $(\mathcal{P}(\kappa),\subseteq)$

Is there a cardinal $\kappa \neq \emptyset$ and a connected poset $P$ of cardinality $\leq \kappa$ such that there is no surjective order-preserving map from $(\mathcal{P}(\kappa),\subseteq)$ onto ...
2
votes
0answers
94 views

Isomorphic subcategories of directed graphs and presets

For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive and ...
5
votes
2answers
128 views

Image of poset with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology $\tau_{\text{int}}(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = ...
3
votes
1answer
102 views

Product of posets with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\leq x\}$ and ...
2
votes
1answer
129 views

Terminology question for maps between posets

Let $P$ and $Q$ be two poset (partially ordered sets) and $\phi : P \to Q$ an order-preserving function. I would like to know whether there is a name and perhaps a different characterizations of such ...
2
votes
1answer
167 views

When do infinitesimals split in dimension groups?

Let $G$ be a dimension group (i.e. a directed, unperforated abelian group satisfying the Riesz interpolation property) with order unit $u\in G^{+}$. There is a canonical positive group homomorphism ...
2
votes
1answer
68 views

Universal and left-factoring order-preserving maps

Trying to get a different angle for the question Fixed points and universal maps for posets, I want to compare universal maps to a different kind of functions. First recall that for posets $P,Q$ an ...
2
votes
0answers
86 views

Pre-Order induced by continuous functions

I'm an newbie in category theory, but I want use it to solve a pre-order question I encountered in my research: Let $X$ be a convex&compact subset of $\mathbb{R}^n$. $f,g: X \rightarrow [0,1]$ ...
10
votes
3answers
225 views

Is the homomorphism poset directed if the codomain is directed?

Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. (We call a function $f:P\to Q$ order preserving if $x\leq y$ in $P$ ...
3
votes
1answer
213 views

When is the homomorphism poset between posets a lattice?

Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. There is a natural ordering relation on $\text{Hom}(P,Q)$ given by $f\leq ...
3
votes
0answers
30 views

Unique representability of bounded distributive lattices

Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space. A poset $(P,\leq)$ is called ...
3
votes
1answer
59 views

Minor ordering for finite graphs

Let $\mathcal{G}_{<\omega}$ be the set of graphs $G = (V,E)$ such that $V = \{0,\ldots,n\}$ for some $n \geq 0$ and $E \subseteq \mathcal{P}_2(V) = \{\{a,b\} : a,b \in V \textrm{ and } a\neq b\}$. ...
1
vote
1answer
118 views

Does the category of posets have pushouts and pullbacks? [duplicate]

Let $\mathbf{Poset}$ be the category of partially ordered sets with order-preserving maps. Does $\mathbf{Poset}$ have both pushouts and pullbacks?
3
votes
1answer
153 views

Uniformizing a relation on ordered sets

Suppose $A$ and $B$ are (complete) ordered sets. Suppose $R\subseteq A\times B$, and $f(a)=\inf\{b : (a,b)\in R\}$ $g(b)=\inf\{a : (a,b)\in R\}$ then what can we call $f$ and $g$? Perhaps there is ...
3
votes
0answers
106 views

Order dimension vs topological dimension of a poset

Let $(P,\leq)$ be a partially ordered set (poset). We define the ordering dimension $\textrm{dim}_\textrm{ord}(P)$ of $(P,\leq)$ to be the smallest cardinal $\kappa$ such that there exist a set of ...
4
votes
3answers
286 views

Classification of countable posets?

Is there a classification of countable posets where between each two comparable elements there is a third element between them?
13
votes
1answer
1k views

Is every continuous function measurable?

This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow. In non-Hausdorff topology it is standard to ...
29
votes
3answers
872 views

Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorem. Suppose $P$ and $Q$ are posets ...
3
votes
1answer
90 views

reduction to np hard ordering problem

I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm. My problem is: I have M auctions and in each auction I have N ...
3
votes
2answers
174 views

Directed subposet of a poset containing the minimal elements

The following appears naturally in a certain context: Let $P$ be a graded partially ordered set. Let $M$ be the subset of minimal elements of $P$. Define subsets $E_i$ inductively as follows: First, ...
3
votes
3answers
113 views

Are there any results on well-quasi-ordering of languages?

There are a number of papers that I can find about well-quasi-orders in formal lnaguage theory, by Kunc, de Luca, D'Alessandro, and Varricchio, among others. I am interested, however, in well-quasi ...
5
votes
0answers
94 views

Mapping graphs to ordinals

Robertson-Seymour theorem implies that graph minor relation is a well-quasi-ordering, which means (among other things) that this relation can be extended to a well-order, and other result says that ...
7
votes
1answer
248 views

The category of categories and adjunctions

What is known about the category that has small categories as objects and adjunctions as morphisms? Obviously, it has neither terminal nor initial objects. But what about other kinds of limits? Are ...
8
votes
0answers
123 views

Can $Ded(\kappa)$ be a supremum?

Definition If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then write $D(\kappa,\lambda)$. $Ded(\kappa)=\sup_\lambda \{D(\kappa,\lambda)\}$. It ...
4
votes
1answer
176 views

Profinite completion of a partial order

In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...
3
votes
2answers
158 views

Finite lattices whose number of join-irreducibles does not exceed its height

In a finite distributive lattice $L$ one has $height(L) = |J(L)|$ i.e. the size of the largest chain equals the number of join-irreducible elements. Briefly, this follows by arranging the subposet ...
4
votes
2answers
264 views

Equality-preserving embeddings of finite trees

For finite trees $T_{1}$ and $T_{2}$ labelled by elements of some infinite set $S$, (we may assume that $S=\mathbb{N}$ without loss of generality), we define an equality-preserving embedding $f$ to be ...
8
votes
1answer
191 views

Status of Barany's conjecture?

One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks: Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$? A ...
0
votes
1answer
206 views

Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$

Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...
4
votes
2answers
164 views

Embedding a linearly ordered free monoid into a linearly ordered group

A linearly ordered (shortly, l.o.) monoid is a triple $\mathbb M = (M, \cdot, \le)$ for which $(M, \cdot)$ is a (multiplicatively written) monoid and $\le$ is a total order on $M$ such that $xy < ...