Questions tagged [lattice-theory]
The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.
421
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Periods of Coxeter transformation associated to root posets
$\DeclareMathOperator\Co{Co}$Let $P$ be the root poset associated to a simple Lie algebra.
Let $L=L(P)$ denote the distributive lattice of order ideals of $P$ and let $\Co_L$ denote the Coxeter matrix ...
7
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Classification of posets that are quotient posets of the Boolean lattice
Quotient posets of the Boolean lattice $B_n$ have interesting properties and are for example discussed in chapter 5 of Stanley's book on algebraic combinatorics.
$B_n/G$ for a subgroup $G$ of the ...
5
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191
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Monster group as automorphism group of a distributive lattice
It is known that every finite group is the automorphism group of a finite distributive lattice.
Question: What is the minimal order of a distributive lattice $L$ such that the automorphism group of $...
4
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0
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121
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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 $C_L:=- U^{-1} U^{T}$ ...
1
vote
0
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111
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Rowmotion for general lattices
Let $L$ be a finite lattice and $x \in L$ with covers $r_1,...,r_l$ in $L$.
One can define $row(x):= \min \{ y | y \leq r_1 \lor \cdots \lor r_l $ and $ y \nleq r_1 \lor \cdots \lor \overline{r_t} \...
2
votes
2
answers
403
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lattice suprema vs pointwise suprema
What is the difference between the lattice supremum and the pointwise supremum of a family of functions? I mean, given a family of real valued functions $\mathcal{F}$, is the function $\sup\mathcal{F}:...
4
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122
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A multiplication for distributive lattices via rowmotion
Let $L$ be a finite distributive lattice. In case $L$ is Boolean one can define a multiplication by $a*b:=(a \cup b) \setminus (a \cap b)=(a \cup b) \cap (a \cap b)^c$, where $(-)^c$ denotes the ...
8
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2
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441
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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 ...
3
votes
1
answer
155
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How to construct a lattice having a subset of a given relations?
I am given a (smallish, say $n=14$ element) set $X$, and a set $R$ of (a few hundred) quadruples of elements $(a, b, c, d)$ with $a,b,c, d\in X$.
I want to construct lattices on $X$, such that for all ...
6
votes
2
answers
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Group structure for distributive lattices
On the (finite) Boolean lattice there is a group structure given by the symmetric difference and this group is an elementary abelian 2-group.
Question: Does there exist a natural group structure on ...
11
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11
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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 ...
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answer
484
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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 ...
3
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0
answers
98
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Shellability and order filters in the partition lattice
Choose $n\in\mathbb N$. Let $B$ be a non-empty subset of $[n]:=\{1,2,\dots,n\}$. Consider the set of partitions of the set $[n]$ with exactly $|B|$ parts such that each part has exactly one member ...
5
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1
answer
436
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Extending submodular functions from a sublattice
This came about when I was studying the connection between matroids and strong
greedoids, but it has broken through into a subject I am not particularly
familiar with: submodular functions on lattices....
18
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1
answer
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Uses of Zorn's Lemma when the thing is actually unique
There is a revised version, which I might substitute for this one, but I would like to keep this as evidence of priority for the "special condition".
Are there uses of the sledgehammer Zorn'...
12
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2
answers
470
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Lattice structure in the root poset
Let $W$ be a Coxeter group with simple generators $s_1$, $s_2$, ..., $s_r$. Let $\Phi^+$ be the corresponding positive root system, with $\alpha_i$ the positive root corresponding to $s_i$. Bjorner ...
2
votes
1
answer
108
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Decreasing sequences in a finitely generated closure algebra
I am interested in finitely generated closure algebras (as a special case of Heyting algebras), and in decreasing sequences of elements within such an algebra that have no lower bound.
Call two ...
10
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2
answers
425
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Class of lattices that excludes $M_3$?
It is well known that a lattice is distributive iff it excludes as a sublattice $N_5$ (the pentagon) and $M_3$ (three unordered elements with a top and bottom). Further, a lattice that only excludes $...
8
votes
1
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The lattice of analogues of Robinson's $Q$
This question was asked and bountied at MSE without response.
Call a sentence $\varphi$ in the language of arithmetic $Q$-like iff $\mathbb{N}\models\varphi$ and $\{\varphi\}$ is essentially ...
0
votes
2
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489
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The union of two cuts is a cut?
Every poset $\langle P, \leq \rangle$ has a Dedekind-MacNeille completion, a complete lattice that embeds $\langle P, \leq \rangle$.
For $A \subseteq P$, the upset $U(A) = \{p \in P\ |\ \forall a \in ...
3
votes
2
answers
121
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Explicit lifting characterization of complete lattices among posets?
It's well-known that the complete lattices are characterized among all posets as the regular-injectives. That is, a poset $L$ is a complete lattice if and only if $L$ has the right lifting property ...
1
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0
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Given a cubic fourfold is it possible to extend an isometry on the primitive cohomology to a complete marking?
Let $X$ be a (smooth) cubic fourfold and $F$ its Fano variety of lines.
It is known that $F$ is an hyperkähler fourfold. Let $(L,u)$ be an abstract lattice with a distinguished element $u$ isomorphic ...
3
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397
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Equivalence of the union-closed sets conjecture that is locally weaker of any use?
Let $F$ be a union-closed family. We call $F$ minimal if for every $x\in \cup(F)$ we find $S\in F$ such that $S\backslash \{x\} \in F$. It is sufficient to proof the union-closed sets conjecture for ...
0
votes
2
answers
99
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Is this ordering on the set of all covers of $\omega$ a (complete) lattice?
Let ${\frak C} \subseteq {\cal P}({\cal P}(\omega))$ be the collection of all covers of $\omega$ (that is, ${\cal C} \in {\frak C}$ iff $\bigcup {\cal C} = \omega$.)
We define the following binary ...
4
votes
1
answer
179
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Distributive lattices and axiom of choice
What form of the axiom of choice is equivalent (in ZF) to the statement that every distributive lattice is isomorphic to a lattice of sets?
3
votes
2
answers
443
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A ‘canonical’ bounded lattice with proper de Morgan negation?
Call a lattice negation $\neg$ proper de Morgan negation iff it satisfies the following conditions.
$\neg\neg a=a$.
$\neg(a\vee b)=\neg a\wedge\neg b$ and $\neg(a\wedge b)=\neg a\vee\neg b$.
$a\...
4
votes
2
answers
534
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Which lattices are quotients of finite powerset lattices?
Let $S$ be a finite set, and let $2^S$ be its powerset, regarded as a lattice. Let $L$ be a quotient (in the category of lattices and maps which preserve $\top,\bot,\wedge,\vee$) of $S$. What can we ...
4
votes
1
answer
277
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Does the lattice of partitions map onto the lattice of subsets?
Let $X\neq \emptyset$ be a set and let $X^X$ denote the collection of all functions $f:X\to X$. We put a binary relation (reflexive and transitive), the composition preorder on $X^X$ by setting for $f,...
4
votes
1
answer
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Upper bound for an expression for distributive lattices
Let $L$ be a finite distributive lattice with minimum $0$ and Maximum $1$ and join-irreducible elements $j_1,...,j_l$ and meet irreducible elements $m_1,...,m_l$.
Let $J_L:= \sum\limits_{i=1}^{l}{| [...
1
vote
1
answer
179
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Lattices from quaternion algebras (MAGMA software)
I am studying the paper "Lattice Packing from Quaternion Algebras" from 2012 about the construction of ideal lattices.
In Section 3.3 the authors construct very interesting examples of lattices using ...
2
votes
2
answers
433
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A formula for a right adjoint in terms of a left
For those familiar with (covariant) Galois connections, you may have noticed that they can be viewed as categorical adjunctions. A Galois connection is a pair of maps between posets $X$ and $Y$
$$ f_{\...
2
votes
0
answers
146
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Is there a name for relations that are compatible with composition and union?
I’m dealing with relations on relations $\mathcal{R} \subseteq \mathcal{P}(A \times A) \times \mathcal{P}(A \times A)$ that have the following properties:
$(R_{1}, S_{1}) \in \mathcal{R} \mathrel\...
4
votes
1
answer
202
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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$...
3
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answers
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Terminology for set systems: "trace" or "projection"?
Although the following question is not in itself mathematical, it is the expertise/breadth of the research community in mathematics that I wish to appeal to, beyond the filtered/trained search results ...
7
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On the isomorphism of the lattices of submodules of certain free modules
Let $K,L$ be two finite extensions of the $p$-adic field
$\mathbb{Q}_p$ of the same degree. Let $\mathcal{O}_K$ and
$\mathcal{O}_L$ be the ring of integers of these two fields, and let
$\mathcal{O}_K^...
0
votes
2
answers
249
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Representations of modular lattices, extension to cellular sheaves
There are various "representation theorems" for lattices such as Birkhoff's Representation Theorem that states that every finite distributive lattice is isomorphic to a quasi-sublattice of the lattice ...
17
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1
answer
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Best introductory texts on pointless topology
As I understand it, there are three canonical textbooks on pointless topology: the classic "Stone Spaces" by Johnstone, "Topology via Logic" by Steve Vickers, and the newer "Frames and Locales" by ...
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1
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Are non-trivial interval-isomorphic posets lattices?
We say that a partially ordered set $(P,\leq)$ is interval-isomorphic if for all $a<b \in P$ we have $P \cong [a,b]$, where $[a,b]=\{x\in P:a\leq x\leq b\}$.
Suppose $(P,\leq)$ is interval-...
6
votes
1
answer
207
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Pairwise non-isomorphic interval-isomorphic lattices
Let us call a lattice $(L,\leq)$ interval-isomorphic if for all $a<b \in L$ we have $L \cong [a,b]$, where $[a,b]=\{x\in L:a\leq x\leq b\}$.
Are there $2^{\aleph_0}$ pairwise non-isomorphic ...
6
votes
2
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940
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Remarkable applications of Dickson's lemma
Dickson's lemma states that, for a fixed $k \in \mathbf N^+$, every set of $k$-tuples of natural numbers has finitely many elements that are minimal with respect to the product order induced on $\...
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vote
1
answer
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From Steiner systems to geometric lattices to matroids
I am looking for a specific matroid. I found a source that claimed to discuss these matroids, but then, only discusses geometric lattice. Even more, in that paper, the geometric lattice that seems to ...
7
votes
1
answer
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Is a sign-preserving operator on $L^2$ a multiplication?
Let $T:L^2(\mu)\to L^2(\mu)$ be a linear and continuous operator, where $L^2(\mu)$ is the (real) $L^2$-space to some $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$.
$T$ is assumed to be sign-...
11
votes
2
answers
374
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When is the poset of acyclic orientations of a graph a lattice?
$\def\inv{\mathrm{inv}}\def\Acyc{\mathrm{Acyc}}$Let $G$ be a graph whose vertices are numbered $\{ 1,2, \ldots, n \}$. Given an orientation $\omega$ of $G$, define the inversions of $\omega$, written $...
1
vote
0
answers
45
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Equivalence relations: Cosimplicial semilattice?
For $n\ge 0$, let $E_n$ be the set of all equivalence relations on $[n]:=\{0,\dotsc,n\}$. Now given two equivalence relations $R,R'\in E_n$, we build their join
$$R\vee R' := \langle R\cup R'\rangle,$$...
3
votes
1
answer
328
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Infinite group generated by a single coset
Let $G$ be an infinite countable group having a core-free subgroup $H$ such that the interval $[H,G]$ in the subgroup lattice $\mathcal{L}(G)$ is ACC of infinite length, and for every $K \in (H,G]$, $...
4
votes
1
answer
264
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Infinite distributive laws in atomless free sigma-algebra
Let $\frak{A}$ be the free $\sigma$-algebra on $\omega_1$ free $\sigma$-generators. Then $\frak{A}$ is not completely distributive because it is atomless. However, is it $\omega$-distributive in the ...
4
votes
1
answer
147
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A condition on minimal restricted subalgebras of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. Assume that the following condition holds:
For every restricted ideal $I$ of $L$, the minimal restricted subalgebras ...
1
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1
answer
179
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What does it mean to be meet dense? [closed]
What does it mean that a set of principal ideals is meet dense in a lattice of all order ideals?
5
votes
1
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201
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Which complete orthocomplemented lattices arise as the lattice of 'regular opens' in a closure space?
Every complete Boolean algebra arises as the lattice of regular open sets in some topological space, namely given a complete Boolean algebra $B$, the corresponding Stone space $S(B)$ will be ...
1
vote
2
answers
184
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Reference request: lower sets of a preorder form a lattice
Consider a set $S$ with a preorder $\preceq$ (a preorder is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $...