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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questions
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Algebraic applications of an order-theoretic idiom of recursion
Many algebraic constructions must surely use the following observation, probably disguised as one of its proofs:
Lemma Let $s:X\to X$ be an endofunction of a poset such that
$X$ has a least element $...
9
votes
1answer
156 views
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 ...
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0answers
19 views
Adjoints to the inclusion of complete Boolean algebras in complete Heyting algebras
Let $\bf{Bool}$ be the full subcategory of $\bf{Hey}$, the category of complete Heyting algebras and meet,joins and implication preserving maps between them.
Is $\bf{Bool}$ a (co)reflective ...
1
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1answer
38 views
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 ...
6
votes
2answers
286 views
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
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1answer
259 views
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
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0answers
48 views
Structure of connected (finite undirected) graphs with respect to homomorphism
After the appropriate quotient, (finite undirected) graphs with homomorphism form a lattice: the meet is the categorical product G x H, the join is the disjoint union G + H. This question concerns the ...
0
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2answers
110 views
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
2answers
100 views
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 ...
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90 views
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 ...
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83 views
Varieties of non-distributive lattices with De Morgan negations and their ‘canonical’ members
After some revisions and checks, I have decided to repost the question.
Recall that the variety $\mathcal{DM}$ of (distributive) De Morgan lattices has the lattice $\mathbf{4}$ as its canonical ...
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331 views
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 ...
1
vote
2answers
71 views
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
1answer
124 views
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
2answers
359 views
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
2answers
361 views
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
1answer
220 views
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
1answer
78 views
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
1answer
112 views
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 ...
1
vote
1answer
103 views
What information about the lattice $\Lambda$ can be recovered from the metric space $\mathbb{R}^n/\Lambda$?
Let $\Lambda\subset\mathbb{R}^n$ a lattice, i.e., a discrete subgroup that spans $\mathbb{R}^n$. Now we can look at the torus $T=\mathbb{R}^n/\Lambda$ which naturally carries the metric $d_T$ induced ...
2
votes
2answers
238 views
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
0answers
130 views
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\...
3
votes
1answer
129 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$...
3
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0answers
83 views
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 ...
6
votes
0answers
70 views
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
1answer
97 views
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 ...
14
votes
1answer
515 views
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 ...
1
vote
1answer
76 views
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
1answer
194 views
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 ...
5
votes
2answers
464 views
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 $\...
1
vote
1answer
135 views
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 ...
6
votes
1answer
138 views
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
1answer
235 views
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 $...
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0answers
36 views
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
1answer
258 views
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
1answer
163 views
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
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0answers
116 views
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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1answer
100 views
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?
4
votes
1answer
85 views
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
2answers
97 views
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 $...
2
votes
0answers
89 views
Open sets on a Stone space
If $B$ is a Boolean algebra (possibly assumed complete), is there a standard name for the Heyting algebra (or frame) $L := \Omega(S(B))$ of open sets on the Stone space $S(B)$ of $B$, — or for the ...
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135 views
Relative strength and propositional indistinguishability of non-distributive lattices
Consider the class of bounded non-distributive lattices $\mathbf{Mn}$ ($n\geqslant 3$).
(from left to right: M3, M4, Mn)
Now consider a propositional language over $\{\wedge,\vee,\neg\}$ with the ...
7
votes
1answer
109 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 ...
3
votes
1answer
306 views
What's concrete model for Coxeter complexes?
We know for every Coxeter system $(W,S)$ there is a Coxeter complex associated by its cosets of parabolic subgroups. In Wachs's note Poset Topology p.12-13 she mentioned for the Coxeter complex of ...
2
votes
0answers
67 views
Distributive generators of a lattice
Given a finite lattice $L$, suppose $L$ is generated by a set $X$ such that the distributive law holds for all $a,b,c\in X$ i.e. $a\lor (b \wedge c) = (a\lor b)\wedge (a \lor c)$.
Is $L$ ...
2
votes
2answers
176 views
Is the intersection of Boolean sublattices a Boolean sublattice?
Let $L$ be a boolean lattice, $A$ and $B$ sublattices of $L$ that are
themselves boolean lattices, and suppose that $I = A \cap B$ is
nonempty.
Is $I$ a boolean sublattice of $L$? Is it a ...
9
votes
1answer
291 views
Fibers of the morphism from the free Heyting algebra to the free Boolean algebra
For $k\in\mathbb{N}$, let $H_k$ be the free Heyting algebra on $k$ variables $p_1,\ldots,p_k$ and $B_k$ be the free Boolean algebra on the same $k$ variables. Thus, $B_k$ has $2^{2^k}$ elements (...
2
votes
2answers
179 views
The space of Borel function modulo comeager sets is Dedekind complete
Let $(X,\tau)$ be a topological space. Denote by $Bor(X)$ the space of Borel functions $f:X\to\mathbb{R}$ where we identify two functions whenever they agree on the complement of a meager set. We ...
4
votes
0answers
329 views
How should the proof of the XYZ theorem be understood?
The XYZ Theorem of Shepp [1] states that for a given poset $P$. Consider the probability space of all the linear extensions of $P$, where each possible extension is equally likely. Then for any three ...
4
votes
0answers
222 views
Generalization of a theorem of Øystein Ore in group theory: the infinite case
This post is the infinite version of this one, and is motivated by an exchange with Carmela Musella and Maria De Falco. We are interested in relative versions of the following Ore's theorem and ...
