# 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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### 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 ...

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### 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 \...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $X\subseteq S$ ...

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### 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 ...

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### 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.
...

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### 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 ...

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### 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 $\eta \mapsto c$ ...

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### 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}(...

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### 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^\...

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### 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, ...

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### 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 $[0,1]$, $[0,1]\cap \mathbb{Q}$...

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### 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 $i^2=a,j^2=b;a,b\in F^\times$.
...

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### 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, ...

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### 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 ...

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### ($\sigma$-)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 ...

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### 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 (“...

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### 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 = \...

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### 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 $...

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### 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 ...

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### 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 ...

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### 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.
...

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### 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 ...

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### 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, ...

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### 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 $\tau$ ...

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### 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 ...

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### 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, \...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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}(\...

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### 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 ...

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### 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, \...

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### 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 ...

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### 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 $f_{i}$ ...

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### 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 ...

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### 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}: |{\...

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### 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 ...

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### "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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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 ...