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0
votes
1answer
183 views

Order-Perserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
6
votes
1answer
86 views

Partial Orders realized by Prime Ideals on commutative rings

Is there a general criterion for which partial orders can be realized by the prime ideals of commutative rings (like we have for topological spaces - https://en.wikipedia.org/wiki/Spectral_space)? ...
1
vote
0answers
90 views

The lattice of graphs under vertex abstractions

I am curious to know if the following structure has been studied, or if anything similar is in the literature. For $n \in \mathbb{N}$, let $G = ([n],E)$ be a digraph. A partition of a subset $V$ of ...
-1
votes
1answer
64 views

Interval topology on $(\mathbb{N}^\mathbb{N},\leq^*)$

Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by $$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$ where $\downarrow x = \{y\in Q: y\leq ...
2
votes
1answer
100 views

Is the interval topology of $(\mathbb{N}^\mathbb{N}, \leq^*)$ connected?

Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by $$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$ where $\downarrow x = \{y\in Q: y\leq ...
5
votes
0answers
112 views

Functoriality of $\mathsf{Cu}$

I have always been happy with the proof of the functoriality of the Cuntz semigroup $\mathsf{Cu}$ given in arXiv:0902.3381, where the isomorphism $$\mathsf{Cu}(A)\cong W(A\otimes K)$$ is used, $A$ ...
4
votes
0answers
75 views

Face structures of chain polytopes

For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain ...
1
vote
1answer
73 views

Transitive reduction from a linear extension of a partial order

Is there an efficient algorithm to create a transitive reduction from a single linear extension of a given partial order? Update: I'm aware of the time complexity of computing a transitive reduction ...
5
votes
0answers
71 views

Minimal Hausdorff topologies compatible with a bunch of functions

Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, ...
1
vote
1answer
324 views

Modern books about orders and algebras on trees [closed]

Please help to find books about orders and algebras on trees. If there is no modern books, please advice good old ones! I'm more interested in finite trees (my current problem), but infinite ones are ...
1
vote
0answers
58 views

Partial orders on tabloids

Let $n \in \mathbb{N}$ and let $\lambda \vdash n$, a partition of $n$. By a $\lambda$-tabloid I mean a row-tabloid of shape $\lambda$. There is a well-known order on the set of $\lambda$-tabloids, ...
11
votes
1answer
530 views

Any further applications of Freudenthal's 1936 Spectral Theorem?

Seemingly completely forgotten, back in 1936, the Dutch mathematician Freudenthal, quite well known at the time, proved his so called Spectral Theorem, see chapter 6 in Luxemburg & Zaanen : Riesz ...
1
vote
0answers
27 views

Form of binary function over poset that is monotone over first and antitone over second argument

if I have a partially ordered set $P$, and I have a function $f: P \times P \to \mathbb{R}$ that is monotone over the first and antitone over the second argument, i.e. for any $a,b,c \in P$ $a ≤ b ...
0
votes
2answers
1k views

Topological sort of partial order into sorted sets

Given a partial order of elements, one can use topological sorting to produce a sorted list of elements. For example, if we have the partial order A->B and A->C, then the possible topological sort ...
8
votes
0answers
101 views

Product of Partial Orders

Define the transpose product of a partial order $P$ over a set $S$ in the following way. The direct product of a partial order $P \subseteq S \times S$ and its converse, $P^{op}$, gives a partial ...
0
votes
1answer
78 views

Prime ideals containing the finite members of ${\cal P}(\omega)$

Let ${\frak P}$ denote the collection of prime ideals containing the finite members of ${\cal P}(\omega)$, and order ${\frak P}$ by set inclusion. What is the cardinality of ${\frak P}$, and what's ...
0
votes
1answer
89 views

Boolean algebras and free filters generated by chains

Suppose $\mathbf{B}$ is a complete Boolean algebra with an infinite domain $B$. Suppose $\mathbf{B}$ is atomic (i.e. every element is the supremum of some set of atoms). This algebra contains the ...
-1
votes
1answer
128 views

Quotients of posets

Let $\mathbf{Poset}$ denote the category of partially ordered sets and order-preserving maps. Does $\mathbf{Poset}$ have quotients?
1
vote
0answers
97 views

Examples of value quantales

In his paper "Quantales and continuity spaces" R. C. Flagg gives the following examples of value quantales: the lattice $\bf{2}$ of truth values with usual addition, the lattice $\mathbb{R}_{+}$ of ...
1
vote
1answer
74 views

“Interval” terminology for (partially) ordered sets

Let $(X,\preceq)$ be a poset. Is there a standard, generally recognised term for a set $A \subset X$ satisfying $$ \forall x,y,z \in X, \ (x \in A \ \textrm{ and } \ z \in A \ \textrm{ ...
4
votes
5answers
736 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 ...
0
votes
0answers
78 views

Properties of a specific antichain of a lattice formed by the cartesian product of finite ordered sets

Introduction Let $X$ be a poset of all $n$-tuples, $x = (x_1, x_2, ..., x_n)$, where $0 \leq x_i \leq m_i - 1$ for $i = 1, ..., n$ together with the relation $x \prec y$ defined so that for ...
1
vote
1answer
69 views

Path-connected Hausdorff interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected and $T_2$. Does this imply that $[0,1]$ order-embeds into $P$? (This is a follow-up ...
1
vote
0answers
76 views

Relationship of ${\cal P}(\omega)/fin$ and ${\cal L}$

Define ${\cal L}$ as in this question: the set of functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ at at ...
18
votes
0answers
570 views

A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions $$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$ Suppose we say that ...
1
vote
0answers
84 views

Properties of Coefficients of Order Polynomials [closed]

I am working on a problem involving determining the order polynomial $\Omega_P(k)$ of a partial order $P$, which counts the number of order-preserving transformations/maps from $P$ to the $k$-chain ...
3
votes
1answer
144 views

Properties of the interval topology of the lattice of functions

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 $\downarrow x = \{y\in P: ...
0
votes
1answer
62 views

Does order-preserving equal continuous? [closed]

Let $P,Q$ be posets and endow them with the interval topology $\tau_i(P)$ and $\tau_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?
3
votes
1answer
89 views

Path-connected interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected. Does this imply that $[0,1]$ order-embeds into $P$?
2
votes
1answer
175 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 ...
2
votes
0answers
98 views

Generalizing disjointness

The following definition generalizes set-theoretic disjointess: Definition 0. (Autonomy). Given a Lawvere theory $\mathsf{T}$, a $\mathsf{T}$-algebra $X$, and an indexed family $S$ of subalgebras ...
2
votes
1answer
105 views

Surjectively rigid partially ordered sets

This question is related to a question recently asked by Joel David Hamkins. Let $(P,\leq)$ be a poset. We call it surjectively rigid if the only order-preserving surjective map $f:P\to P$ is the ...
1
vote
2answers
116 views

Is there a name for a partial order in which there is a countable chain which “dominates” the whole space?

Is there a name for a partial order $\preceq$ on a set $X$ with the following property: "there exists a countable set $S \subset X$ such that for all $x \in X$ there exists $y \in S$ with $x \preceq ...
-1
votes
2answers
128 views

Complete non-isomorphic lattices with injective complete homomorphisms between them?

Are there complete lattices $L, K$ such that $L\not\cong K$; there are injective complete lattice homomorphisms $i:L\to K$ and $j: K\to L$ ?
0
votes
1answer
82 views

Does ${\cal Id}(L) \cong {\cal Id}(K)$ imply $L\cong K$?

For any lattice $L$ we denote the complete lattice of the ideals of $L$ by ${\cal Id}(L)$. Are there non-isomorphic lattices $L\not \cong K$ such that ${\cal Id}(L) \cong {\cal Id}(K)$?
1
vote
1answer
53 views

Simplyfing join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_L f(S).$ Is the ...
0
votes
2answers
117 views

Join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_L f(S).$ Suppose ...
8
votes
2answers
291 views

Posets isomorphic to their endomorphism poset

Let $(P,\leq)$ be a poset. We set $$\text{End}(P)=\{f: P\to P: f\text{ is order-preserving}\}$$ and order $\text{End}(P)$ pointwise. Is there a poset with more than 1 point such that $P\cong ...
-1
votes
1answer
58 views

Lattice homomorphism from ${\cal Id}(L)$ onto $L$

For any lattice $L$ we denote the complete lattice of the ideals of $L$ by ${\cal Id}(L)$. If $L$ is complete, is there a lattice homomorphism from ${\cal Id}(L)$ onto $L$?
0
votes
1answer
117 views

Spliting of short exact exact sequences of partially ordered groups

Consider a short exact sequence of partially ordered groups $$0 \longrightarrow H \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} G/H \longrightarrow 0 ,$$ where $H$ is a ...
3
votes
1answer
94 views

Incomplete lattice homomorphisms between complete lattices (2)

Let $L, K$ be complete lattices. A lattice homomorphism $f: L\to K$ is said to be incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_K f(S).$ Consider the ...
0
votes
1answer
107 views

Incomplete lattice homomorphisms between complete lattices

Let $L, K$ be complete lattices. A lattice homomorphism $f: L\to K$ is said to be incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_K f(S).$ Suppose that ...
0
votes
0answers
49 views

Antichains in subgroups up to group automorphism

A family of subgroups of a group G will denote a nonempty collection of subgroups closed under conjugation and further passing to subgroups. In a Noetherian group (any ascending chain of subgroups ...
1
vote
2answers
149 views

Embedding finite lattices into the lattice of partitions of a finite set

For any set $X$ we denote by $\text{Part}(X)$ the set of all partitions of $X$, ordered by the refinement ordering. It is well known that this is a complete lattice for all sets $X$. Let $L$ be a ...
8
votes
1answer
135 views

Optimal strategy for game of 'online sorting' into a poset

Consider a single-player game played with an arbitrary finite poset, and a random number generator with a known distribution: Each turn, the RNG produces a number, and the player must assign that ...
12
votes
1answer
343 views

How long can a cycle of antichains in a finite partial order be?

Suppose that $X$ is a finite partially ordered set. Then a subset $A\subseteq X$ is said to be an antichain if there do not exist elements $a,b\in A$ with $a<b$. Let $\mathcal{A}_{X}$ be the set of ...
2
votes
1answer
140 views

Finitely generated ordered monoids and noetherian subsets

(This question was asked a long time ago on MSE but got no answer so far...) Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...
13
votes
2answers
477 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 ...
0
votes
1answer
82 views

Dedekind-MacNeille completion of the strictly increasing members of $\omega^\omega$

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ ordered by $f\leq g$ iff $f(n) \leq g(n)$ for all $n\in \omega$. Set $K = \{f\in \omega^\omega: m<n\in \omega \implies ...
3
votes
1answer
165 views

Order dimension of $\omega^\omega/(fin)$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we say $f\simeq g$ if and only if $\exists N \in \omega$ such that $f(n) = g(n)$ for all $n\geq N$. ...