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The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.

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107 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
65 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 ...
2
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1answer
71 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
137 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 ...
2
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1answer
151 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....
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0answers
52 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 ...
3
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1answer
98 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 ...
2
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1answer
82 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
71 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
33 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 ...
2
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0answers
50 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 ...
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0answers
32 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
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1answer
49 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
45 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
331 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
157 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
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1answer
175 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 ...
4
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0answers
64 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
81 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
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2answers
225 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 ...
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2answers
81 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
322 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
66 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
208 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?
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2answers
203 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 ...
6
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1answer
249 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
107 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,...
15
votes
2answers
319 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
129 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
47 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 ...
2
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0answers
39 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
133 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
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1answer
153 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
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2answers
461 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
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1answer
85 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
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1answer
199 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
177 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 ${\...
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1answer
138 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
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1answer
223 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 ...
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1answer
105 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 $y$,...
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0answers
211 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
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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 ...
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1answer
40 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
214 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 ...
1
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1answer
108 views

Is an Eulerian subgroup lattice boolean?

Let $G$ be a finite group and $\mu$ the Möbius function of the subgroup lattice $\mathcal{L}(G)$. The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K<G, \ g \...
4
votes
1answer
298 views

Existence of a non-Eulerian atomistic lattice with this property on the Möbius function

Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$. Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},...
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0answers
97 views

Bound on number of nxn grids with lexicographical ordering / poset structure

Given $n\in\mathbb{N}$, consider the numbers $\{1,\ldots,n^2\}$ and a permutation $\pi\in S_{n^2}$. It induces pairs $(1,\pi(1))$, $\ldots$, $(n^2,\pi(n^2))$. Consider an $n\times n$ grid. How many ...
0
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1answer
68 views

Definition of an Orlicz modular space

In Nowak (1989), a modular $\rho$ on a vector lattice is defined by the following properties (N1) $\rho(x)=0\implies x=0$; (N2) $\lvert x\rvert \le \lvert y\rvert\implies \rho(x) \le \rho(y)$; (N3) ...
2
votes
1answer
39 views

Equations satisfied by finite modular lattices

I found this very interesting paper by Freese, The Variety of Modular Lattices is Not Generated by its Finite Members, which shows that finite modular lattices satisfy an identity that is not ...
2
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1answer
102 views

Atomicity of blocks in a Hilbert lattice

Where can I find the proof that any block (maximal boolean subalgebra) $\mathbf{B}$ of the orthomodular lattice $\mathcal{L}$ of closed subspaces of a separable Hilbert space $\mathcal{H}$ is atomic?