Questions tagged [lattice-theory]
The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.
395
questions
5
votes
2
answers
322
views
Generalization of the concept of a measure
Consider the following generalization of the concept of a measure:
Let $L = (X, \lor, \land, \bot)$ be a semi-bounded lattice.
Let $M = (Y, \bullet, e)$ be a commutative monoid.
An $(L, M)$-measure is ...
- 1,721
5
votes
0
answers
177
views
Is this "trimming" of a supersolvable semimodular lattice known?
Let $L$ be a finite (upper) semimodular lattice. Recall that this means $L$ is graded and its rank function $\rho\colon L \to \mathbb{N}$ satisfies
$$ \rho(x) + \rho(y) \geq \rho(x\vee y)+\rho(x \...
- 21.2k
13
votes
1
answer
235
views
Can any poset of cardinality $\leq 2^{\aleph_0}$ be embedded in ${\cal P}(\omega)/(\text{fin})$?
We endow ${\cal P}(\omega)$ with an equivalence relation by saying that $A\simeq_{\text{fin}} B$ iff the symmetric difference $A\Delta B$ is finite. The resulting set of equivalence classes is denoted ...
- 43.6k
2
votes
0
answers
268
views
Does this monoid have a name?
Fix a positive integer $n \geq 1$. Let $M$ be the monoid with generators $S=\{x_0,x_1,\ldots,x_n\}$ and relations $R = \{ \alpha x_0 = \beta x_0\colon \alpha,\beta \in S^*, |\alpha|=|\beta|\}$, where $...
- 21.2k
2
votes
0
answers
66
views
Lattices where complete joins/meets are countable joins/meets
I am looking for information, in particular references, on the following lattice-theoretical property:
L is a complete lattice; for every uncountable subset S of L, there is a countable subset C of S ...
2
votes
1
answer
155
views
Parametrization of topological algebraic objects
There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...
- 5,189
0
votes
0
answers
42
views
Proving a property in De Morgan residuated lattices
A residuated lattice is an algebra $(L, \wedge, \vee,\odot, \rightarrow, 0, 1)$
of type $(2, 2, 2, 2, 0, 0)$ satisfying the following axioms:
(RL1) $(L, \wedge, \vee)$ is a bounded lattice (the ...
- 1
3
votes
1
answer
127
views
Why Is Pudlak's relation on the family of one- or two-element subsets of a set transitive?
The following comes from Definition 2 in Pavel Pudlak, "A new proof of the congruence lattice representation theorem," Algebra Universalis 6 (1976), 269-275.
Let $X$ be a set. Let $F$ be a ...
- 1,198
4
votes
1
answer
249
views
Is every finite poset a subset of a finite complemented distributive lattice?
Let $(X,\succeq)$ be a poset. I have the following two questions:
Is it true that there exists a finite complemented distributive lattice (a Boolean lattice) $(S, \succeq^*)$ such that $X\subseteq S$ ...
- 63
2
votes
1
answer
139
views
Defining states on von Neumann algebras from filters on the projection lattices
Let $M$ be a von Neumann algebra, $P(M)$ be its projection lattice, and $\mathcal{F}$ a proper filter on $P(M)$. Does there exist a state $\varphi$ (not necessarily normal) s.t. $\varphi(p) = 1$ for ...
- 732
11
votes
1
answer
324
views
Lattices of clones: is 4 worse than 3?
Let $\mathscr{C}_n$ be the lattice of clones on the $n$-element set $\{1,...,n\}$. $\mathscr{C}_2$ is complicated but countable, but $\mathscr{C}_3$ (and all higher lattices) is of size continuum.
...
- 18.1k
0
votes
0
answers
29
views
When can we separate two pairs in ${\mathbb H}_n$, although it is not a lattice?
Recall that a lattice is a partially ordered set $E$ for which any pair $a,b\in E$ admits a least upper bound and a greatest lower bound. Remark that given four elements $a_i,b_j$ ($j=1,2$), in order ...
- 50.2k
10
votes
1
answer
650
views
Are modular lattices shallow?
Let $A$ be a universal algebra with finitely many finitary operations. Write $F_n$ for the $n$-ary operations.
We define the affine maps on $A$ inductively: $\eta \mapsto \eta$ and $\eta \mapsto c$ ...
- 6,162
1
vote
0
answers
43
views
Is a “well-behaved” closed subbasis for the topology generated by a closure operator a closed basis for the closure operator itself?
Let $\Omega$ be a set, $\mathcal{c}: \mathcal{P}(\Omega) \rightarrow \mathcal{P}(\Omega)$ be a closure operator (i.e., $\mathcal{c}$ satisfies $X \subseteq \mathcal{c}(X)$ and $\mathcal{c}(\mathcal{c}(...
- 732
1
vote
0
answers
53
views
Must an ultrafilter orthogonal to all ultrafilters containing an element $a$ contains $a^\perp$?
Let $L$ be an orthocomplemented lattice. We may consider the collection $U$ of ultrafilters on $L$. We say two elements $a, b \in L$ are orthogonal to each other, written as $a \perp b$, if $a \leq b^\...
- 732
0
votes
0
answers
73
views
Is a principal filter in a free Heyting algebra a projective Heyting algebra?
A Heyting algebra is a bounded distributive lattice $(L,\vee,\wedge,0,1)$ together with a binary operation $\rightarrow$ called implication or relative pseudocomplementation with the property that, ...
- 1,198
6
votes
1
answer
342
views
Is every homogeneous poset a lattice?
A poset $(P,\leq)$ is homogeneous if $P\cong [a,b]$ for all $a,b\in P$ with $a<b$ (where $[a,b] := \{x\in P: a\leq x\leq b\}$).
Examples of homogeneous posets include $[0,1]$, $[0,1]\cap \mathbb{Q}$...
- 43.6k
4
votes
1
answer
431
views
A note on orders in quaternion algebras
Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$.
...
- 163
4
votes
1
answer
270
views
Boolean algebra of the lattice of subspaces of a vector space?
Recall that a Boolean algebra is a complemented distributive lattice. The set of subspaces of a vector space comes very close to being a boolean algebra. It satisfies all the required properties, ...
- 929
2
votes
0
answers
48
views
What is known about sublocales defined by regular nuclei?
(For basic terminology, which is supposed to be standard anyway, see this other question, which inspired this one.)
I am interested in nuclei $j\colon L\to L$ on a frame $L$ which are regular elements ...
- 27.8k
0
votes
0
answers
50
views
($\sigma$-)completeness of Riesz spaces
Let $E$ be a Riesz space (or vector lattice). I am searching for examples (possibly with references) of especially $\sigma$-complete Riesz spaces (each countable set of positive elements has a ...
3
votes
1
answer
178
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 (“...
- 27.8k
5
votes
2
answers
471
views
Do germs of open sets around a point form a frame?
Let $X$ be a topological space and $x \in X$ a point. Let $\Omega$ be the set of open sets (viꝫ. the topology) of $X$, and $\Omega_x$ the set of germs around $x$ of open sets, that is, $\Omega_x = \...
- 27.8k
1
vote
0
answers
46
views
Something like Dedekind-MacNeille completion
The Dedekind–Macneille completion of a poset $P$ can be represented as a complete lattice $\widehat P$ consisting of all lower subsets (order ideals) $I$ for which
$I=(I^\uparrow)^\downarrow$, where $...
- 837
4
votes
1
answer
144
views
How large must algebras with a given congruence lattice be?
This is a follow-up to a recent question of mine:
For $n\in\mathbb{N}$ let $C(n)$ be the smallest $k$ such that every bounded lattice with cardinality $\le n$ which is isomorphic to the congruence ...
- 18.1k
8
votes
1
answer
317
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 ...
- 18.1k
4
votes
1
answer
245
views
When is this topology compatible with the partial ordering?
This question was first asked here, on math stack exchange, but wasn't able to attract any attention. Now that I am thinking more, it feels like the most suitable place for this question is here.
...
- 929
7
votes
0
answers
515
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 ...
- 7,733
1
vote
2
answers
133
views
Lattices formed by unions of elements in an antichain
Let $A_1, \dots, A_k$ be incomparable subsets (of $\{1, \dots, n\}$) and consider the poset $P$ consisting of all possible unions of these under inclusion. Its not hard to see that this is a lattice, ...
- 43
14
votes
1
answer
245
views
Is there a countably infinite closed interval in the lattice of topologies?
Is there an interval of the form $[\sigma,\tau]$ in the lattice of topologies on some set $X$ such that $|[\sigma,\tau]| = \aleph_0$?
In other words, do there exist two topologies $\sigma$ and $\tau$ ...
- 16.9k
2
votes
0
answers
119
views
Proof of Crapo's complementation theorem
In Crapo's work "Möbius inversion in lattices," he gave a second proof of his complementation theorem:
$$ \mu(0,1) = \sum_{x, y \in s^\perp} \mu(0,x) \zeta(x, y) \mu(y,1)$$
where $s$ is an ...
- 101
12
votes
2
answers
766
views
Ultrafilter lemma for arbitrary lattice
Can someone kindly confirm whether the ultrafilter lemma for arbitrary (i.e., not necessarily Boolean) bounded lattices is equivalent to Zorn's lemma?
To be precise, if $\mathbf{L} = (L, \leq, \land, \...
- 273
7
votes
1
answer
177
views
Which lattices have non-trivial linear representations?
Suppose we have a a bounded lattice $L$. We might ask: does there exist a non-trivial linear representation of $L$, i.e. a lattice homomorphism $\rho: L \to \text{Sp}(V)$, where $V$ is a non-zero ...
- 101
7
votes
1
answer
174
views
Free median algebras and maximal linked systems
$\DeclareMathOperator\MLS{MLS}$Recall that the median operation, on the power set $2^Y$ of subsets of a set $Y$, is the ternary law $m(A,B,C)$ mapping a triple of subsets to the set of elements ...
- 57.7k
11
votes
1
answer
625
views
Determining whether a lattice is the face lattice of a polytope - NP hard or undecidable?
According to this source (p. 10), determining whether a simplicial complex is a simplicial sphere (the sphere recognition problem) is undecidable.
According to this source, determining whether a ...
- 11.7k
2
votes
0
answers
110
views
Product-decomposition of ${\cal P}(\omega)/\rm{fin}$ [closed]
For $A,B\in {\cal P}(\omega)$ let us say that $A\simeq_{\rm{fin}} B$ if both $A\setminus B$ and $B\setminus A$ are finite. It is easy to see that this establishes an equivalence relation on ${\cal P}(\...
- 43.6k
2
votes
1
answer
135
views
Reference for lattices as algebraic structures
I want to study lattices as a structure related to ring theory. I am familiar with lattices as a beginner but I want to go further and know their connections to ring theory. Do you know a book which ...
- 33
6
votes
1
answer
139
views
Variable elimination for propositional formulas in Heyting algebras
By an (intuitionistic) propositional formula $\varphi(x_1,\ldots,x_n)$ I mean a formula built up from a (finite) number of variables $x_1,\ldots,x_n$ using connectors $\top, \bot, \land, \lor, \...
- 27.8k
4
votes
0
answers
145
views
To whom is the classification of atomic, modular finite lattices due?
Here lattice means a poset with meets and joins. A lattice is called atomic if every element is a join of atoms. There are a few different ways to define modular for finite lattices: one is that the ...
- 21.2k
2
votes
0
answers
50
views
Can we decompose an increasing net of functions into two increasing nets with prescribed supports?
Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...
- 5,189
10
votes
4
answers
880
views
Are arbitrary nonempty intersections of principal filters principal?
Suppose $\langle L,\leq\rangle$ is a lattice with join $\sqcup$. Let $F_1$ and $F_2$ be principal filters on $L$. Thus, for $i\in I=\{1,2\}$ there are $x_i\in L$ so that $F_i=\{y\in L:x_i\leq y\}$.
In ...
- 335
0
votes
1
answer
149
views
How to find the closest point given the Voronoi relevant vectors?
Let M be the generator matrix of a $N\times N$ lattice, and $\tilde{N}$ the set of Voronoi relevant vectors. The Voronoi cell for the origin can be written as $\text{Vor}_{\bf 0}(M)=\left\{{\bf x}: |{\...
- 163
1
vote
0
answers
54
views
Consequences of having unbounded points in a bornology
For a set $X$ a bornology $\mathcal{B}$ is essentially an ideal in the power set $\mathcal{P}(X)$. Many sources including Wikipedia state additional property that $X = \bigcup \mathcal{B}$. Call it a ...
- 377
5
votes
0
answers
164
views
"Cluster algebra" structure for finite distributive lattices
Let $P$ be an $n$-element poset and $J(P)$ the distributive lattice of its order ideals (i.e., the downwards-closed sets).
For each $I\in J(P)$ let $x_I \in \mathbb{R}^{n}$ be the indicator function ...
- 21.2k
6
votes
0
answers
132
views
Natural bijection between join- and meet-irreducibles in modular lattices?
A well known property of finite modular lattices is that they have the same number of join-irreducible and meet-irreducible elements. I was wondering if there exists a natural bijection between these ...
- 3,370
6
votes
1
answer
248
views
Distributive lattice of subspaces
Let $V$ be a finite dimensional vector space. Let $\Lambda$ be a collection of subspaces of $V$ such that, if $X$ and $Y$ are in $\Lambda$, then $X\cap Y$ and $X+Y$ are in $\Lambda$. This makes $\...
- 148k
2
votes
0
answers
126
views
Counting integer partitions below some Young diagram
Question: Given positive, coprime integers $m<n$, consider the Young diagram $Y$ formed by the lattice points in the Cartesian plane lying below the line from $(0,0)$ to $(m,n)$ and within the ...
- 705
14
votes
5
answers
748
views
Birkhoff's representation theorem vs matroid-geometric lattice correspondence
This question is motivated by the superficial observation that Birkhoff's representation theorem and the cryptomorphism between matroids and geometric lattices are sort of similar. The former says ...
- 3,370
0
votes
0
answers
111
views
Colimits in the category of suplattices
I want to compute coequalizers in the category $\mathcal{S}up$ of complete lattices and $\bigvee$-preserving maps. One way (I think?) is to use the dual equivalence
$$
\mathcal{Sup} \leftrightarrows \...
- 127
2
votes
1
answer
209
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 ...
- 1,826