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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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 ...
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Fixed points for finitary distributive lattices bijection

Birkhoff's Fundamental Theorem of Finite Distributive Lattices says that there is a bijection $$ \{ \textrm{finite posets}\} \to \{ \textrm{finite distributive lattices}\} $$ $$ P \mapsto J(P), $$ ...
Sam Hopkins's user avatar
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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 ...
Pedram's user avatar
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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&...
Sam Hopkins's user avatar
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"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(...
Dominic van der Zypen's user avatar
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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 ...
Dale's user avatar
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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 ...
David Gao's user avatar
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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 ...
Dominic van der Zypen's user avatar
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Reference Request for "Finite Semilattice with Top and Bottom is a Lattice"

Let $\mathcal{O}(P)$ be a finite, completely distributive lattice of all lower sets ordered by set inclusion. Moreover, let $K =\; \mathrel{\{} h(x) \mathrel{|} x \in \mathcal{O}(P) \mathrel{\}}$ be ...
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The projective structure of a non-distributive modular lattice

Thrall states ("On the Projective Structure of a Modular Lattice", 1951, https://scholar.google.com/scholar?cluster=4998496641867146321): One measure of the complexity of a modular lattice ...
Dale's user avatar
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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 ...
user76284's user avatar
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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 \...
Sam Hopkins's user avatar
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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 ...
Dominic van der Zypen's user avatar
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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 $...
Sam Hopkins's user avatar
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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 ...
Christian Ronse's user avatar
2 votes
1 answer
201 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 ...
erz's user avatar
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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 ...
Arenna's user avatar
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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 ...
Tri's user avatar
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4 votes
1 answer
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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 $X\subseteq S$ ...
Pedram's user avatar
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1 answer
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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 ...
David Gao's user avatar
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11 votes
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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. ...
Noah Schweber's user avatar
1 vote
0 answers
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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 ...
Denis Serre's user avatar
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10 votes
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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 $\eta \mapsto c$ ...
Ville Salo's user avatar
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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}(...
David Gao's user avatar
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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^\...
David Gao's user avatar
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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, ...
Tri's user avatar
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6 votes
1 answer
349 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 $[0,1]$, $[0,1]\cap \mathbb{Q}$...
Dominic van der Zypen's user avatar
5 votes
2 answers
609 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 $i^2=a,j^2=b;a,b\in F^\times$. ...
Hussein Eid's user avatar
4 votes
1 answer
378 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, ...
Bumblebee's user avatar
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2 votes
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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 ...
Gro-Tsen's user avatar
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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 ...
Mwenya Alagona's user avatar
3 votes
1 answer
205 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 (“...
Gro-Tsen's user avatar
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5 votes
2 answers
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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 = \...
Gro-Tsen's user avatar
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1 vote
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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 $...
Gejza Jenča's user avatar
4 votes
1 answer
149 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 ...
Noah Schweber's user avatar
8 votes
1 answer
332 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 ...
Noah Schweber's user avatar
4 votes
1 answer
279 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. ...
Bumblebee's user avatar
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7 votes
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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 ...
Paul Taylor's user avatar
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1 vote
2 answers
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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, ...
Moty Katzman's user avatar
14 votes
1 answer
252 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 $\tau$ ...
Will Brian's user avatar
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2 votes
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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 ...
yaoliang's user avatar
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12 votes
2 answers
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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, \...
Menander I's user avatar
7 votes
1 answer
189 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 ...
Yegreg's user avatar
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7 votes
1 answer
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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 ...
YCor's user avatar
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11 votes
1 answer
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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 ...
M. Winter's user avatar
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2 votes
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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}(\...
Dominic van der Zypen's user avatar
2 votes
1 answer
147 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 ...
13571's user avatar
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6 votes
1 answer
160 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, \...
Gro-Tsen's user avatar
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4 votes
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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 ...
Sam Hopkins's user avatar
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2 votes
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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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