The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.

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Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
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0answers
18 views

Characterization of Dedekind complete Riesz spaces by strictly positive functionals

I was browsing throughout the literature and I found the following fact: Every $\sigma$-Dedekind complete Riesz space $E$ which admits a strictly positive functional $f$ is Dedekind complete. I ...
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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 ...
3
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2answers
89 views

About Kurosch-Ore theorem

Where can I find the proof of Kurosch-Ore theorem in lattice theory? The statement of this theorem is: Let $L$ be a modular lattice with $0$ and $1$ that satisfies both chain conditions. Then for any ...
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1answer
159 views

Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?

Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and ...
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61 views

What are universal abstract $\sigma$-algebras on $\sigma$-frames?

Originally asked on MSE. In this paper, the authors make the following definitions: An (abstract) $\sigma$-algebra is a boolean algebra with countable joins. A $\sigma$-frame is a bounded lattice ...
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1answer
166 views

“Zorn's Lemma guarantees that all algebraic frames are spatial.” Why?

In at least two papers (here and here) Jorge Martínez and Eric R. Zenk say that Zorn's Lemma implies that all algebraic frames are spatial. However, I haven't been able to find an actual explanation ...
2
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1answer
71 views

Is this a sufficient condition for distributivity of a lattice?

I asked this question here on Math.SE but uptil now it was not answered. So I decided to give it a try here. Thank you in advance. If a lattice $L$ is distributive then it can be shown that for ...
3
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1answer
143 views

Can a length n distributive lattice be embedded into Bn?

Let $\mathcal{L}$ be a finite distributive lattice, then it is known that it can be embedded into a finite boolean lattice (see theorem 8.5. p91 in this note). Let $n$ be the length of ...
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94 views

A finite distributive lattice which may be represented as the normal subgroup lattice of a supersolvable group

Is there a supersolvable group $G$ with the lattice of all its normal subgroups, order-isommorphic to the 18-element lattice of down-sets of this poset: ? It has been proved that not every finite ...
0
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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 ...
2
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3answers
107 views

Non-principal prime ideals in infinite distributive lattices

Given an infinite distributive lattice $L$, does $L$ contain a non-principal prime ideal $I$, or a non-principal prime filter $F$? ($I$ is said to be principal if there is $x\in L$ such that $I=\{y\in ...
5
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1answer
108 views

An equivalent definition for the adiamond lattices

A lattice is called adiamond if it admits no sublattice equivalent to the diamond lattice $M_3$ below: The top interval of a lattice is the interval between the meet of all the maximal elements and ...
2
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2answers
96 views

Are lattices quotients of their Dedekind-MacNeille completion?

Let $L$ be a lattice and let $\textbf{DM}(\cdot)$ denote the Dedekind-MacNeille completion. Is there a lattice $L$ that is not a quotient of $\textbf{DM}(L)$? And what if we generalise this question ...
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3answers
576 views

How is the free modular lattice on 3 generators related to 8-dimensional space?

Here are three facts which sound potentially related. What are the actual relationships? In 1900, Dedekind constructed the free modular lattice on 3 generators as a sublattice of the lattice of ...
9
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176 views

Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion of finite groups. Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ ...
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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 ...
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79 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 ...
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62 views

Lattice Flatness Measure

I am looking for the definition of a flatness measure in lattice theory. More generally, I am looking at finite-height lattices and I want to measure their complexity, with a perfectly flat lattice ...
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58 views

Standard name / symbol for intersection in Brouwerian lattices

A Brouwerian lattice has a lower adjoint $\cdot - B$ to $B\lor\cdot$. It is called pseudodifference. The main reference is http://www.jstor.org/stable/1969038 Once you have pseudodifference, you can ...
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61 views

Is there an official name for the intersection of the join-irreducible representations of two lattice elements?

Given a lattice provided with a join-irreducible representation of its elements, there is a natural "intersection" operator $A \mathbin{\dot\cap} B$ that returns the join of the setwise intersection ...
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77 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 ...
3
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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: ...
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1answer
76 views

Is there a surjective lattice homomorphism $f: {\cal L}\to \mathbb{N}^\mathbb{N}$?

Let ${\cal L}$ be defined as in this question. Is there a surjective lattice homomorphism $f: {\cal L}\to \mathbb{N}^\mathbb{N}$, where $\mathbb{N}^\mathbb{N}$ is the set of all functions, ordered ...
11
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1answer
479 views

How big is the lattice of all functions?

Define the lattice $(\mathcal{L},\prec)$ as the set of all function $f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ ...
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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 $j: K\to L$ ?
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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 ${\cal Id}(L) \cong {\cal Id}(K)$?
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1answer
116 views

Cardinality of pairwise non-isomorphic complete lattices on an infinite cardinal $\kappa$

Suppose $\kappa$ is an infinite cardinal. Let $\cal S$ be a collection of pairwise non-isomorphic complete lattices on the ground set $\kappa$. What cardinality can $\cal S$ have at most? Is the ...
3
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0answers
69 views

Hausdorff spaces with lattice isomorphism between the topologies [closed]

For $i=1,2$ let $(X_i,\tau_i)$ be Hausdorff spaces such that the lattices $\tau_1, \tau_2$ are isomorphic. Does this imply that $(X_1, \tau_1) \cong (X_2,\tau_2)$? (This is a follow-up question to ...
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1answer
52 views

$T_2$-spaces such that the lattices of open sets can be embedded into each other

Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$. Does this imply that $(X,\tau)\cong (Y,\sigma)$?
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1answer
26 views

Reducing join-incomplete lattice homomorphisms to homomorphisms with co-domain ${\bf 2}$

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 $f(\bigvee_L S) > \bigvee_L f(S).$ Is it true ...
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65 views

An equality for the trace of the join of a non-degenerate indecomposable system of projections in a finite factor

Let $M \subset B(H)$ be a finite factor (see for example here p2, or there) with a trace $tr$. The subset of projections of $M$ is naturally a lattice, noted $(\mathcal{P}(M), \wedge, \vee)$. A ...
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1answer
143 views

Is the top interval of a finite distributive lattice an hypercube lattice?

Let $(L,\wedge,\vee)$ be a finite distributive lattice. Let $M$ be the (unique) maximum element. An element $a \in L$ is called maximal if $a \le a' < M$ implies $a = a'$. Let $b = ...
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28 views

Characterization of complete lattices with 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 $f(\bigvee_L S) > \bigvee_L f(S).$ How can ...
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1answer
62 views

Contracting 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 $f(\bigvee_L S) \neq \bigvee_L f(S).$ If $f:L\to ...
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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 ...
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1answer
125 views

Lattices without prime ideals

If $\kappa$ is an infinite cardinal, is there a lattice $L$ of cardinality $\kappa$ such that $L$ contains no prime ideals?
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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 $L$?
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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 ...
3
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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 ...
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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 ...
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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 ...
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1answer
166 views

The groups with symmetric subgroups lattice

Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice. If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the ...
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1answer
72 views

Set of ideals of the set of finite subsets of $\mathbb{N}$

In any lattice $L$, an ideal is a subset $I\subseteq L$ that is downward closed, and moreover $a,b\in I$ implies $a\vee b\in I$. We denote by ${\cal I}(L)$ the set of ideals of $L$, ordered by set ...
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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 ...
4
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1answer
151 views

Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every $x′>x$ and $y′>y$, we have \begin{equation*} f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′). \end{equation*} Suppose $f$ and $g$ are supermodular, ...
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1answer
493 views

Generalization of the fundamental theorem of cyclic groups

Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows: Theorem: $G$ is cyclic iff it admits no two different subgroups with the same order. proof: see ...
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359 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 ...
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131 views

A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height

Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.) ...
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266 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 ...