# Questions tagged [lattice-theory]

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

285
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

**1**answer

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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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 ...

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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 ...

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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 $...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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**1**answer

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?

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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\...

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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 ...

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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,...

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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}{| [...

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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 ...

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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 ...

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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_{\...

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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\...

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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 $y \in Y$...

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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 ...

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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^...

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**1**answer

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 ...

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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 ...

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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-...

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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 ...

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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 $\...

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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 ...

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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-...

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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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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
$$R\vee R' := \langle R\cup R'\rangle,$$...

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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 $K \in (H,G]$, $...

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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 ...

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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 ...

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**1**answer

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?

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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 ...

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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 $...

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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 ...

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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 ...

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**1**answer

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 ...

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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 $L$ ...

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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 ...

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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 (...

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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 ...

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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 ...

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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 ...