Questions tagged [lattice-theory]

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

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Giving $\mathit{Top}(X,Y)$ an appropriate topology

$\DeclareMathOperator\Top{\mathit{Top}}$I am not sure if its OK to ask this question here. Let $\Top$ be the category of topological spaces. Let $X,Y$ be objects in $\Top$. Let $F:\mathbb{I}\...
Amr's user avatar
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12 votes
1 answer
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A dual version of a theorem of Øystein Ore in group theory

This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: Let $[H, G]$ be a distributive interval of finite groups. Then $\exists g \in G$ such ...
Sebastien Palcoux's user avatar
0 votes
2 answers
197 views

Products of maximal inclusions of finite groups with a non-obvious intermediate

Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups. Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
Sebastien Palcoux's user avatar
80 votes
5 answers
6k views

How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. ...
Joel David Hamkins's user avatar
22 votes
0 answers
1k views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
William DeMeo's user avatar
6 votes
2 answers
864 views

Type III factor representation

Does there exist any theorem which permits, under suitable hypotheses, to represent a particular complete orthomodular lattice as the projection lattice of a Type III von Neumann factor?
moppio89's user avatar
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14 votes
2 answers
751 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 ...
Jan-Christoph Schlage-Puchta's user avatar
11 votes
11 answers
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Lattices on classical combinatorial families

I am asking for examples of lattices defined on classical combinatorial families, such as Permutations, Catalan objects, set partitions or integer partitions, graphs. I am mosty interested in lattices ...
Martin Rubey's user avatar
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6 votes
1 answer
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Positivity of the alternating sum of indices for boolean interval of finite groups

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice. Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$. Let the alternative sum ...
Sebastien Palcoux's user avatar
4 votes
1 answer
288 views

Is the top interval of a finite distributive lattice, a boolean lattice?

Let $(L,\wedge,\vee)$ be a finite distributive lattice, and let $1$ its greatest element. An element $a \in L$ is called maximal if $a \le a' < 1$ implies $a = a'$. Let $b$ be the meet of all the ...
Sebastien Palcoux's user avatar
3 votes
1 answer
312 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: y\...
Dominic van der Zypen's user avatar
50 votes
7 answers
14k views

Good lattice theory books?

A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - ...
19 votes
4 answers
1k views

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 ...
Martin Brandenburg's user avatar
10 votes
2 answers
702 views

Can any finite lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$. Question: Can any finite lattice be realized as ...
Sebastien Palcoux's user avatar
10 votes
0 answers
307 views

Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality

$\newcommand\rank[1]{\lvert#1\rvert}$Let $\Bbb{P}$ be a 1-differential poset with a unique bottom element $\emptyset \in \Bbb{P}$. With some minor abuse in terminology, The Plancherel measure state $...
Jeanne Scott's user avatar
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8 votes
2 answers
441 views

Which complete orthomodular lattices arise from von Neumann algebras?

Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice. Question 1: Is the construction $A \mapsto \Pi(A)$ a ...
Tim Campion's user avatar
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7 votes
1 answer
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Enumerative characterisation of boolean lattices II

This is a sequel of this post. The boolean lattice $B_n$ is graded with rank numbers $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, and $n2^{n-1}$ edges. Question: Is a graded lattice with the ...
Sebastien Palcoux's user avatar
4 votes
4 answers
471 views

What are the rank 3 boolean intervals [H,G], with G simple group?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$. The lattice $B_{3}$ is the following: Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $...
Sebastien Palcoux's user avatar
4 votes
1 answer
202 views

Map on class of all finite posets coming from maximal sized antichains

Let $P$ be a finite poset. Let $\mathcal{A}$ denote the set of antichains of $P$. Equip $\mathcal{A}$ with a partial order $\preceq$ whereby $X \preceq Y$ means for all $x \in X$ there exists $y \in Y$...
Sam Hopkins's user avatar
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3 votes
1 answer
206 views

Computing the Heyting operation on the frame of nuclei

(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound (“...
Gro-Tsen's user avatar
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2 votes
1 answer
150 views

What is the smallest partition lattice PART(M) containing the lattice P(N) of subsets of a finite set of N elements

It's been known for 35 years that every finite lattice can be embedded in a finite partition lattice (Pudlak and Tuma, Algebra Universalis 1980, Volume 10, Issue 1, pp. 74--95). I don't follow the ...
user43467's user avatar
2 votes
1 answer
140 views

Efficiently embedding finite Boolean algebras into lattices of set partitions?

Let $P_n$ be the lattice of set partitions of $[n] = \{1,2,\dots,n\}$, let $B_n$ be the Boolean algebra of subsets of $[n]$. Is there some $n_0$ such that for all $n \ge n_0$ it is possible to ...
András Salamon's user avatar
1 vote
0 answers
223 views

Does $[H_i , G_i]$ distributive imply $[H_1 \times H_2, G_1 \times G_2]$ modular?

Let $L(G)$ be the subgroup lattice of $G$ and $[H, G]$ an interval in $L(G)$. A lattice $(L, \wedge, \vee)$ is distributive if $a∨(b∧c) = (a∨b) ∧ (a∨c)$, $\forall a,b,c \in L $, and is modular if ...
Sebastien Palcoux's user avatar
20 votes
3 answers
2k views

Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way: Let $X$ be a set and let $\mathcal{P}(...
Bipolar Minds's user avatar
15 votes
2 answers
676 views

Why do the projections in the Calkin algebra not form a lattice?

Let $H$ be an infinite dimensional separable complex Hilbert space. Denote by $\mathcal{B}(H)$ the C*-algebra of bounded operators on $H$, $\mathcal{K}(H)$ the ideal of compact operators on $H$, and $\...
Iian Smythe's user avatar
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14 votes
1 answer
312 views

Are these two quotients of $\omega^\omega$ isomorphic?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say $f\simeq_{\text{fin}} g$ if there is $n\in \omega$ such that $f(k) = g(k)$ for all $k\geq n$. ...
Dominic van der Zypen's user avatar
13 votes
5 answers
3k views

Online introduction to Lattice Theory?

Apart from J. B Nation's Notes on Lattice Theory, is there any other (mostly introductory) material on Lattices available online? NB: The last update of Nation's notes was 2017, as of Feb 2023.
adamo's user avatar
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12 votes
1 answer
443 views

How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of Garrett Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using ...
coudy's user avatar
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10 votes
1 answer
484 views

is there a ‘nice’ lattice on the set of unlabelled graphs with $n$ vertices?

It is easy to endow the set of vertex-labelled graphs with $n$ vertices with a lattice structure: take the union and the intersection of the edge set as meet and join respectively. However, I wonder ...
Martin Rubey's user avatar
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8 votes
1 answer
333 views

Example of trickiness of finite lattice representation problem?

I'm trying to come up with a good explanation for my students of why the finite lattice representation problem is difficult. I've already shown that the "greedy approach" to representing the ...
Noah Schweber's user avatar
8 votes
3 answers
636 views

Homotopy type of some lattices with top and bottom removed

The only reaction to this question on math.SE was 21 views in 4 days, so I decided to repost it here. I am not changing anything. There was an interesting question on MO which OP removed by some ...
მამუკა ჯიბლაძე's user avatar
8 votes
3 answers
1k views

The set of complements equal to the complement of set

Consider $A \subset \{0,1\}^n$ I want $A$ to have two properties. $1.$ $A$ is increasing, i.e., If $x \in A$ and $x \subseteq y$ then $y \in A$ too. [$x \subseteq y$ means that every coordinate ...
MR_BD's user avatar
  • 550
8 votes
1 answer
235 views

Finite posets for which all intervals are atomic

Let $P$ be a finite poset which is a lattice with $0,1 \in P$. An atom in $P$ is an upper cover of $0$ and a coatom is a lower cover of $1$. $P$ is atomic if every element is a join of atoms and ...
Christian Stump's user avatar
7 votes
0 answers
582 views

A new and subtle order-theoretic fixed point theorem

Sometimes a very simple argument appears out of the blue and overturns a subject. It is not based on pre-existing theory and heavy involvement in such a theory is actually a handicap in finding such ...
Paul Taylor's user avatar
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7 votes
0 answers
245 views

Interesting uniform posets

A sequence $(P_0,P_1,\ldots)$ of finite posets is called uniform if: each $P_n$ is graded of rank $n$ with a minimum $\hat{0}_n$ and a maximum $\hat{1}_n$; for any $p \in P_n$ with $\mathrm{rank}(p)=n-...
Sam Hopkins's user avatar
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7 votes
0 answers
337 views

A homological algebra approach to the Union-closed sets conjecture

I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...
Mare's user avatar
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6 votes
4 answers
1k views

Is every Heyting algebra the Lindenbaum algebra of an intuitionistic first order theory?

This question comes after the comments in the recent related question Sigma-complete Lindenbaum algebras?, but in its current form is sufficiently different in my opinion, and so I decided to follow ...
godelian's user avatar
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6 votes
1 answer
672 views

A meet-semilattice with top element that is not a lattice?

I am reading Francis Borceux’s “Handbook of Categorical Algebra I” and on page 135 it says In particular a finite version of 4.2.5 does not hold: a finitely complete and well-powered category ...
tgeng's user avatar
  • 163
6 votes
2 answers
1k views

Embedding a Brouwerian lattice into a Boolean lattice

I have already asked a similar question at https://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra but have received no answer. Sorry, I ask a ...
porton's user avatar
  • 739
6 votes
2 answers
453 views

Divisibility labeling on a boolean lattice and positive Euler totient

Let $B_n$ be the rank $n$ boolean lattice (i.e. the subset lattice of $\{1,2, \dots , n \}$). Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum of $B_n$. Let $f: B_n \to \mathbb{N}$ be a ...
Sebastien Palcoux's user avatar
6 votes
1 answer
609 views

To what extent can a von Neumann algebra be determined by its projection lattice structure?

Let $ M, N $ be von Neumann algebras, $ P $ (resp. $Q$) the projection lattice of $M$ (resp. $N$). Any isomorphism $ \varphi : M \to N $ on the level of involutive algebras induces an isomorphism $ \...
Rick Sternbach's user avatar
6 votes
1 answer
212 views

Fixed points for finitary distributive lattices bijection

Birkhoff's Fundamental Theorem of Finite Distributive Lattices says that there is a bijection $$ \{ \textrm{finite posets}\} \to \{ \textrm{finite distributive lattices}\} $$ $$ P \mapsto J(P), $$ ...
Sam Hopkins's user avatar
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6 votes
1 answer
2k 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 ...
Sebastien Palcoux's user avatar
5 votes
1 answer
253 views

Divisibility labeling on a boolean lattice and nonzero Euler totient

Let $B_n$ be the subset lattice of $\{1,2, \dots , n \}$, also called the boolean lattice of rank $n$. A labeling $f: B_n \to \mathbb{N}_{\ge 1}$ is called acceptable if $\forall a,b \in B_n$: ...
Sebastien Palcoux's user avatar
5 votes
1 answer
243 views

Enumerative characterisation of boolean lattices

The boolean lattice of rank $n$ (noted $B_n$) is the subset lattice of $\{1,2, \dots , n \}$. See the Hasse diagram of $B_3$ below: The Hasse diagram of $B_n$ is of length $n$, with $2^n$ vertices ...
Sebastien Palcoux's user avatar
5 votes
0 answers
162 views

(When) is the Dedekind-MacNeille completion of a po-set Hausdorff?

Let $X$ be a p.o. Consider the topology on $X$ generated by $$U_{x}^{-}:=X\setminus (x\uparrow),\quad U_{x}^{+}:=X\setminus (x\downarrow), \quad x\in X$$ Throughout this discussion I shall refer to ...
Thomas's user avatar
  • 263
4 votes
0 answers
121 views

Distributive lattices with periodic Coxeter matrix

Let $L$ be a finite distributive lattice and $U$ its incidence matrix with entries $u_{i,j}=1$ iff $i \leq j$ and $u_{i,j}=0$ else. Then $U^{-1}$ is the Moebius matrix of $L$ and $C_L:=- U^{-1} U^{T}$ ...
Mare's user avatar
  • 25.8k
4 votes
1 answer
249 views

Constructing ordered fields with lattice structure from ordered fields without lattice structure, and vice versa, in constructive mathematics

This post originated from my reference request for the definition of an ordered field in constructive mathematics: Proper definition of ordered field in constructive mathematics We are working in ...
Madeleine Birchfield's user avatar
4 votes
0 answers
408 views

Can infinite bounded distibutive lattices be "arbitrarily wide"?

I was always thinking, in an informal way, that the powerset lattices ${\cal P}(X)$ (where $X$ is an infinite set) are the "widest" bounded distributive lattices with respect to their height. (In ${\...
Dominic van der Zypen's user avatar
4 votes
1 answer
371 views

Existence of a non-Eulerian atomistic lattice with this property on the Möbius function

Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$. Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},...
Sebastien Palcoux's user avatar