Questions tagged [lattice-theory]
The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.
30 questions from the last 365 days
13
votes
2
answers
1k
views
What's the deal with De Morgan algebras and Kleene algebras?
The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is ...
- 32.5k
8
votes
0
answers
244
views
Strengthening of Frankl's union-closed sets conjecture: An algebraic approach
Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: There exists $k\in [n]$ such that:
$$\sum_{k\in A,A\in \mathcal F}\...
- 1,462
2
votes
0
answers
69
views
Link between Carathéodory's criterion and commutation in an orthomodular lattice?
In the theory of outer measures, Carathéodory's criterion constructs from an outer measure $\mu^*$ on $X$ a $\sigma$-algebra $\Sigma$ of subsets of $X$, on which the restriction of $\mu^*$ is a ...
- 193
4
votes
1
answer
189
views
Do idempotents in an abelian category constitute a lattice?
Consider an object $X$ in an abelian category $\mathcal{C}$. We define $\text{Idem}_{\mathcal{C}}(X)$ as the set of idempotent endomorphisms $a$ in $\text{End}_{\mathcal{C}}(X)$, meaning that $a \circ ...
1
vote
0
answers
98
views
Simplicial complexes on $[n] := \{0,\ldots,n\}$ that are identical under any contraction of consecutive vertices
For $n\in\mathbb{N}$, let us denote by $\Omega(n)$ the set of all (possibly empty) “abstract” simplicial complexes on $[n] := \{0,\ldots,n\}$ (“on $[n]$” means “labeled by the elements of $[n]$”). To ...
- 32.5k
9
votes
1
answer
625
views
The reals: a topological lattice in more than the obvious way?
Define a topological lattice as a (not necessarily bounded) lattice in $\textbf{Top}$, i.e. meet and join are continuous maps $X^2 \rightarrow X$. There are two obvious topological lattice structures ...
- 591
6
votes
0
answers
164
views
Can one naturally transform Tamari lattices into distributive lattices with the same number of elements?
Many of the zillions of combinatorial objects counted by Catalan numbers come with various lattice structures.
The $n$th Tamari lattice $T_n$, as originally defined, is the lattice of all those maps $...
- 18.4k
6
votes
1
answer
170
views
Do projections in an $AW^\ast$-algebra form an orthomodular lattice?
I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
- 2,830
5
votes
0
answers
137
views
Looking for a certain finite lattice
I don't think it actually exists, and it should be difficult proving that it doesn't (some background here), but is it possible to build a finite lattice $L$ where the only meet-irreducible elements ...
- 1,000
4
votes
1
answer
181
views
Remove an edge from the Hasse diagram of a finite lattice
If we remove an edge from the Hasse diagram of a finite lattice, as long as any vertex maintain at least one upward edge and at least one downward edge, do we still always have a lattice from the ...
- 1,000
7
votes
1
answer
134
views
Universally closed implies proper for locales
It is well known that:
Theorem.
For a locale (resp. topological space) $X$, the following are equivalent:
$X$ is compact, i.e. every open cover of $X$ has a finite subcover.
For every locale (resp. ...
- 15.9k
6
votes
1
answer
335
views
Existence of pairwise quasi-complementary but not complementary subspaces
Let $𝑋$ be an infinite-dimensional Banach space (complex or real). A subspace of $𝑋$ means a closed linear submanifold. Subspaces $M$ and $N$ of $X$ are quasi-complementary if $M\cap N=\{0\}$ and $M+...
- 391
2
votes
1
answer
118
views
Reference request for a proof of the fact that every congruence-permutable variety is semidegenerate
Given an algebra $\mathbf{A}$, a pair of congruences
$ \alpha ,\beta \in Con(\mathbf {A})$ are said to permute when
$ \alpha \circ \beta =\beta \circ \alpha$, and an algebra
$\mathbf{A}$ is called ...
- 21
11
votes
2
answers
558
views
Whether an isotone bijection from a power set lattice to another sends singletons to singletons
By the work of Paul Cohen (on the continuum hypothesis), one can neither prove nor disprove from the axioms of ZFC that a bijection $f$ from the power set $\mathcal{P}(S)$ of a set $S$ to the power ...
- 10.5k
1
vote
0
answers
66
views
Extending homeomorphisms on closure spaces
Let $C$ be an infinite $T_1$ closure space, which is not a topological space. Suppose $C$ has the exchange property: for $x,y\in C$ and $A\subseteq C$
$$
\big( x\notin\overline{A}, \hspace{4mm} x\in \...
- 2,605
0
votes
1
answer
234
views
Minimum number of elements needed to represent a lattice with a union-closed family of sets
I know that it is possible to represent every finite lattice $L$ with a union-closed family $\mathcal{F}$ containing the empty set: for every $x\in L$, let $S_x=\{y\in L\, :\, y\not\geq x\}$ and $\...
- 1,000
2
votes
0
answers
109
views
Sublattice isomorphisms
Let $L$ be a non-distributive complete (bounded) lattice with the greatest element $1$ and the smallest element $0$.
Definitions: For a given $x,y\in L$, let's use the notation $[x,y] := \{z\in L: x\...
- 2,605
3
votes
0
answers
188
views
Frankl's conjecture for infinite lattices
Given a poset $L$, call it trivial if $\left|L\right| < 2$ and let $\mathcal I\left(L\right)$ be its poset of ideals, $\mathcal C\left(L\right)$ be its set of chains, and $\mathcal M\left(L\right)$ ...
- 31
6
votes
2
answers
502
views
Is a finite lattice determined by its Hasse diagram (as a graph)?
If finite lattices $L_1,L_2$ have Hasse diagrams that are isomorphic as undirected graphs, must $L_1$ and $L_2$ be isomorphic?
NOTE: Sam Hopkins points out that the answer is “no” because there are ...
- 19.7k
3
votes
1
answer
119
views
Non-isomorphic $T_0$-spaces with order-isomorphic topologies
Are there non-isomorphic $T_0$-spaces $(X_i, \tau_i)$ for $i = 1,2$ such that $\tau_1 \cong \tau_2$ when considered as partially ordered sets?
- 48.9k
2
votes
2
answers
271
views
Is every finite lattice isomorphic to a union-closed family of sets containing $\emptyset$?
If a family of sets $\mathcal{F} \subseteq 2^E$ is union-closed and contains $\emptyset$, then $\mathcal{F}$ forms a lattice under the set-inclusion order. To see this, note that unions give the join ...
- 145
4
votes
1
answer
254
views
Are there atoms in the lattice of intermediate logics?
A few days ago I stumbled upon this question on MS. The question is: Does the lattice of intermediate logics have an atom, i.e. an element that is strictly stronger than IPC but not strictly stronger ...
- 143
29
votes
0
answers
665
views
A conjecture about inclusion–exclusion
$\newcommand\calF{\mathcal{F}}
\def\cupdot {\stackrel{\bullet}{\cup}}
\def\minusdot {\stackrel{\bullet}{\setminus}}$This post presents a conjecture that we have with some colleagues. It is about ...
- 391
6
votes
1
answer
254
views
Fixed points for finitary distributive lattices bijection
Birkhoff's Fundamental Theorem of Finite Distributive Lattices says that there is a bijection
$$ \{ \textrm{finite posets}\} \to \{ \textrm{finite distributive lattices}\} $$
$$ P \mapsto J(P), $$
...
- 24.2k
3
votes
1
answer
222
views
Embedding of a poset with "desirable" characteristics
Let $(P, \succeq)$ be a poset (you can assume $\succeq$ is not empty). I am interested in finding a poset $(X, \succeq^*)$ that embeds $(P, \succeq)$, and $\forall x,y \in X$ both of the following ...
- 97
4
votes
1
answer
216
views
Lattice description of matroid duality
Apologies for this very basic question in matroid theory, but I could not find anything about it online after a bit of searching.
There is a well-known bijective correspondence ("cryptomorphism&...
- 24.2k
0
votes
0
answers
117
views
"Infima" and "suprema" in the homomorphism preorder on hypergraphs on $\omega$
$\newcommand{Po}{{\cal P}(\omega)}$
$\newcommand{lh}{\leq_{\text{hom}}}$
If $H_i = (V_i, E_i)$ are hypergraphs for $i = 1,2$, then a map $f:V_1 \to V_2$ is said to be a (hypergraph) homomorphism if $f(...
- 48.9k
4
votes
0
answers
125
views
Can we extend "every finite lattice is a sublattice of partitions of a finite set" to linear and/or finitary lattices?
Pudlák and Tůma https://link.springer.com/article/10.1007/BF02482893 proved that every finite lattice can be
embedded as a sublattice of the partition lattice of a finite set.
Can this be generalized ...
- 429
1
vote
0
answers
80
views
Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$
This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...
- 2,830
9
votes
1
answer
364
views
Are the Boolean algebras ${\cal P}(\omega)/(\text{fin})$ and ${\cal P}(\omega)/(\text{thin})$ isomorphic?
A set $A\subseteq \omega$ is said to be thin if $$\lim\sup_{n\to\infty}\frac{|A\cap \{0,\ldots, n\}|}{n+1} = 0.$$
We say for $A, B\subseteq \omega$ that $A\simeq_\text{fin} B$ if the symmetric ...
- 48.9k