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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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2answers
106 views

Infima and suprema in the “transfer” function ordering

Let $X,Y$ be sets, $f, g:X\to Y$ be functions. We say $u:Y\to Y$ is a transfer function for $g$ to $f$ if $$f = u \circ g.$$ In that case we write $f \leq_t g$. Let $\mathrm{Fct}(X,Y)$ denote the ...
1
vote
0answers
43 views

When does a collection of sets forming a geometric lattice give the flats of a matroid?

Say we have a matroid on a finite set $X$. The collection of its flats forms a geometric lattice under $\subseteq$, where the join is given by intersection. This question is about the converse to ...
2
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1answer
192 views

Triangular conjecture (that implies the Frankl conjecture)

Let $M$ be a $n\times n$ triangular matrix, that entries are $0$ and $1$ , and such that diagonal entries are $1$. A row or a column will be said to be small, if its numer of $1$ is at most $(n+1)/2$. ...
3
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0answers
62 views

To what extend can a von Neumann algebra be determined by its projection lattice structure?

Let $ M, N $ be von Neumann algebras, $ P $ (resp. $Q$) the projection lattice of $M$ (resp. $N$). Any isomorphism $ \varphi : M \to N $ on the level of involutive algebras induces an isomorphism $ \...
5
votes
1answer
177 views

Products, coproducts and equalizers in category of lattices

Background: Let $\mathbf{Lat}$ be the 2-category of lattices which can be viewed as a subcategory of the 2-cateogry of posets $\mathbf{Pos}$, that is, objects in $\mathbf{Pos}$ that have all finite ...
7
votes
1answer
120 views

Is the lattice generated by finitely many subspaces in a finite-dimensional vector space finite?

Let $V$ be a finite-dimensional vector space, let $U_1,\dots,U_n$ be subspaces, and let $L$ be the lattice they generate; namely, the smallest collection of subspaces containing the $U_i$ and closed ...
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2answers
80 views

Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?

Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be regular open if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A ...
1
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1answer
722 views

The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)

Original question (without additional information from Wendy): Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way: Taking the E8 as {128,...
2
votes
1answer
89 views

Gale order on multisets of elements of a lattice

The question Let $L$ be a lattice (in the sense of combinatorics, not number theory). An $L$-bag will mean a finite multiset of elements of $L$. Given an $L$-bag $A$, we consider three possible ...
6
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3answers
145 views

Groups whose poset of direct factors are lattices

Let $G$ be a finite group. Denote by $\mathcal{N}(G)$ the modular lattice of normal subgroups of $G$ and denote by $\mathcal{D}(G)$ the subposet of $\mathcal{N}(G)$ whose elements are the direct ...
4
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1answer
190 views

What is to Stone space of the free sigma-algebra on countably many generators?

I asked the question on MSE. https://math.stackexchange.com/questions/2898377/what-is-the-stone-space-of-the-free-sigma-algebra-on-countably-many-generators The answer I got, however, seems disputed....
3
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0answers
55 views

Is there a name for this kind of structure? (Not quite a lattice-ordered group)

I'm looking at a certain class of groups $G$ that come with a partial order $\le$ on the elements. So far it looks like $(G,\le)$ has the following properties: The partial order is invariant under ...
4
votes
1answer
103 views

Is the free modular lattice linear?

Dedekind proved that the free modular lattice on 3 generators is realisable by the intersections and sums of 4-dimensional subspaces in 8-space. Birkhoff showed that the free lattice is infinite if it ...
3
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1answer
107 views

Is every complete bounded finite lattice equivalent to a sublattice of a powerset lattice?

More precisely, if I have a complete bounded finite lattice $C$, can I compute a lattice-operation-preserving map $C \to P(S)$? for some $S$. If not, is there another universal lattice structure that ...
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0answers
77 views

Homomorphisms between non-isomorphic finite lattices

Let $L_{1}$ and $L_{2}$ be non-isomorphic finite lattices of the same cardinality. Can there exist any lattice homomorphisms between $L_{1}$ and $L_{2}$?
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0answers
34 views

Algorithm of constructing the graph for a partial order set [closed]

Given a finite partial order set $(P,\leq)$. Is there an algorithm for constructing its graph, where, from bottom to top, the ordering goes up? Namely, I want to construct the directed graph ...
3
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0answers
52 views

(When) is the Dedekind-MacNeille completion of a po-set Hausdorff?

Let $X$ be a p.o. Consider the topology on $X$ generated by $$U_{x}^{-}:=X\setminus (x\uparrow),\quad U_{x}^{+}:=X\setminus (x\downarrow), \quad x\in X$$ Throughout this discussion I shall refer to ...
3
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0answers
86 views

A largest lattice of a given Vapnik-Chervonekis dimension

Prove (or disprove) that a largest lattice of Vapnik-Chervonekis dimension at most $k$ which has at most $n\cdot k$ join-irreducible and $n\cdot k$ meet-irreducible elements is the distributive ...
0
votes
1answer
51 views

Complements in $\text{Sub}(\text{Sym}(\omega))$

For any group $G$, we let $\text{Sub}(G)$ be the complete lattice of subgroups of $G$. Let $\text{Sym}(\omega)$ be the group of all bijections $f:\omega\to\omega$. What is an element of $U\in\text{...
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0answers
46 views

getting one tower from two (stronger hypothesis than a previous question with same title)

Suppose that $(L,\leq_L,0,1)$ is a Boolean algebra that is dense as an order (i.e. if $a<_L b\in L$ then there exists $x\in L$, s.t. $a<_L x<_L b$) s.t all non trivial closed segments are ...
2
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1answer
332 views

getting one tower from two

Suppose that $(L,\leq_L,0,1)$ is a distributive and complemented Lattice that is dense as an order (i.e. if $a<_L b\in L$ then there exists $x\in L$, s.t. $a<_L x<_L b$) Suppose that there ...
2
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1answer
161 views

Lattices that top is the top of its join-irreducibles, such that a random element is almost surely greater then any given join-irreducible

Let $(L,\leq,0,1)$ be a lattice, and let's denote by $JI(L)$ the set of its join-irreducibles (i.e. elements that are not the lowest grater bound of two other elements). We suppose that $\sup JI(L)=...
8
votes
1answer
194 views

Automorphisms of power set lattice mod finite

Let $N$ be a countably infinite set and let $\mathcal P$ denote power set. I get that the automorphisms of $(\mathcal P(N),\subseteq)$ are all induced by permutations of $N$. But what can be said ...
5
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0answers
78 views

Maximal subgroups of infinite index and profinite completion

Preliminary remark: I'm mainly interested in an answer (or link to ressources) in the specific context of the first Grigorchuk group, but I believe that it may be of some interest to state the ...
2
votes
1answer
94 views

Boolean completion of a partially ordered set

Given a poset $(P, \leq)$, is there a complete Boolean lattice $B$ and an order-preserving map $i_P: P\to B$ such that for any complete Boolean lattice $B'$ and order-preserving map $f: P\to B'$ ...
6
votes
2answers
236 views

Is an Eulerian lattice shellable?

The notion of Eulerian lattice generalizes the notion of face lattice of a convex polytope. (Bruggesse-Mani): The boundary complex of a convex polytope is shellable. (Björner-Wachs): A poset is ...
4
votes
2answers
92 views

Order-embedding, but no lattice embedding between distributive lattices

Let $L$ be the power set lattice ${\cal P}(\{0,1,2\})$. It is clear that there is an order-preserving injective map from $M_3$ into $L$, but no injective lattice homomorphism (because $L$ is ...
9
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3answers
352 views

Does the lattice of all topologies embed into the lattice of $T_1$-topologies?

Let $\kappa$ be an infinite cardinal, and let $\text{Top}(\kappa)$ be the lattice of all topologies on $\kappa$, ordered by $\subseteq$. Let $\text{Top}^{T_1}(\kappa)$ be the lattice of all $T_1$-...
6
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0answers
69 views

Representing meet-semilattices with vector spaces of specified dimensions

Take $K$ to be a field and take $L$ to be a finite meet-semilattice. I'm interested in the set of functions $n: L \rightarrow \mathbb{Z}^{\ge 0}$ such that there is some function $V$ from $L$ to ...
4
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1answer
212 views

Finite distributive lattices as lattice of ideals of a finite ring

Is there a finite distributive lattice that is not isomorphic to the lattice of ideals of a finite ring?
6
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2answers
218 views

Finite lattice representation problem checking

[Grätzer and Schmidt 1963] proves that every algebraic lattice is isomorphic to the congruence lattice of a universal algebra. A finite lattice is algebraic. The finite lattice representation problem ...
7
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1answer
326 views

Structure of $Hom(L_1,L_2)$, where $L_i$ are distributive lattices

Is there known structures/ or has there been studies on $Hom(L_1,L_2)$ of distributive lattices? Could it be made into a lattice naturally? Is there any structure on the set of ring valued functions $...
3
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1answer
115 views

Dimension of vector spaces as a measure

Let $X$ be a set with a collection of subsets $\mathcal{A}$. A finitely additive measure on $(X, \mathcal{A})$ is a function $\mu: \mathcal{A} \to \mathbb{R}_{\geq 0}$ such that for any two subsets $A,...
16
votes
2answers
323 views

Is the order on repeated exponentiation the Dyck order?

The Catalan numbers $C_n$ count both the Dyck paths of length $2n$, and the ways to associate $n$ repeated applications of a binary operation. We call the latter magma expressions; we will ...
2
votes
2answers
133 views

Möbius function on atomistic (modular) lattices

I have read in a paper that $\mu(0,1)\neq 0$, for the Möbius function $\mu$ on an atomistic (finite, modular) lattice with $0$ and $1$. Could someone provide me with a reference for this claim? Many ...
2
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0answers
48 views

Efficiently embedding finite Boolean algebras into lattices of set partitions?

Let $P_n$ be the lattice of set partitions of $[n] = \{1,2,\dots,n\}$, let $B_n$ be the Boolean algebra of subsets of $[n]$. Is there some $n_0$ such that for all $n \ge n_0$ it is possible to ...
3
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0answers
46 views

Word problem for finitely presented bounded lattices

There is a solution to the word problem for finitely presented (non-bounded) lattices, as well as a solution to the word problem for free bounded lattices. I am assuming that there is a solution to ...
0
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2answers
140 views

Is ${\cal P}(\omega)/\mathrm{(fin)}$ order-isomorphic to its intervals?

Let $a, b \in {\cal P}(\omega)/\mathrm{(fin)}$ with $a<b$. Do we have ${\cal P}(\omega)/(fin)\cong [a,b]$?
3
votes
1answer
162 views

Order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$

Is there an order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$? Or from ${\cal P}(\omega)/(fin)$ onto $[0,1]\cap \mathbb{Q}$?
11
votes
2answers
477 views

What is known about ideal and divisibility lattices of GCD domains and their generalizations?

The divisibility relation "$a$ divides $b$", or concisely, $a \vert b$ defined over a commutative integral domain $R$ with identity induces a partial order on the multiplicative semigroup $R/R^{\times}...
2
votes
1answer
89 views

Is $({\cal P}(\omega), \leq_{\text{inj}})$ a distributive lattice?

For $A, B \subseteq \omega$ we set $A \leq_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a down-set of $B$. It is easy to see that $\leq_{...
3
votes
1answer
201 views

Infima in the Rudin-Keisler ordering

Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by $${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:...
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0answers
208 views

Can infinite bounded distibutive lattices be “arbitrarily wide”?

I was always thinking, in an informal way, that the powerset lattices ${\cal P}(X)$ (where $X$ is an infinite set) are the "widest" bounded distributive lattices with respect to their height. (In ${\...
2
votes
1answer
153 views

Chains of maximum cardinality in distributive lattices

It's quite straightforward to construct a (complete) lattice in which no chain has maximum cardinality: for each $n\in \omega\setminus\{0\}$ let $C_n$ be a copy of $n$ with the chain ordering ...
3
votes
1answer
225 views

Antichains of maximum cardinality: posets vs lattices

The following construction gives a poset such that no antichain has maximum cardinality: For $n\in\mathbb{N}\setminus\{0\}$, let "layer" $n$ consist of an antichain of $n$ points, and as for the ...
-1
votes
1answer
108 views

Covering property of complete distributive lattices

Let $(L,\land,\lor)$ be a complete distributive lattice. Given $x\neq y \in L$, is there a finite set ${\cal I}$ of closed intervals in $L$ such that no member of ${\cal I}$ contains both $x$ and $y$,...
5
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0answers
226 views

What does first-order co-intuitionistic logic look like (and does it have an equivalent type theory)?

So, this is where I'm at so far: Heyting algebras model propositional intuitionistic logic (IL) so do Cartesian closed categories which also model the simply typed lamda calculus co-Heyting algebras ...
2
votes
1answer
76 views

Hausdorff interval topology on distributive lattices

Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\leq ...
0
votes
1answer
44 views

Order-preserving surjections on the Dedekind MacNeille completion

Suppose $L$ is a complete lattice, $P$ is a poset, and $f: L \to P$ is a surjective order-preserving map. If ${\bf DM}(P)$ is the Dedekind MacNeille completion of $P$, is there necessarily a ...
3
votes
2answers
223 views

Order-preserving surjection ${\mathbb N}^{\mathbb N}\to [0,\infty)$

This is kind of a continuation of a recent (closed) question. Is there an order-preserving surjective function $f:{\mathbb N}^{\mathbb N}\to [0,\infty)$ (where for $a,b\in {\mathbb N}^{\mathbb N}$ we ...