Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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

0
votes
0answers
77 views

Tridiagonal conjecture (that implies the Frankl conjecture)

Let $M$ be a $n\times n$ tridiagonal 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
votes
0answers
56 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
131 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
116 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 ...
1
vote
2answers
77 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
votes
1answer
510 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
83 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
votes
3answers
144 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 ...
3
votes
1answer
176 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
votes
0answers
53 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
102 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
votes
1answer
96 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 ...
0
votes
0answers
75 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 $L_{2}$?
1
vote
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 ...
3
votes
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
votes
0answers
85 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{...
1
vote
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
votes
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
votes
1answer
159 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
190 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
votes
0answers
73 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
91 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
232 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
87 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
votes
3answers
329 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
votes
0answers
68 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
votes
1answer
210 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
votes
2answers
211 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
votes
1answer
300 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
votes
1answer
111 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
320 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
votes
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
votes
0answers
45 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
votes
2answers
138 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
161 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
466 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
88 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:...
4
votes
0answers
192 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
149 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
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 ...
-1
votes
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 $y$,...
5
votes
0answers
218 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
43 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
218 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
vote
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
299 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},...