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

learn more… | top users | synonyms

1
vote
1answer
28 views

Homotopy type of some lattices with top and bottom removed

The only reaction to this question on math.SE was 21 views in 4 days, so I decided to repost it here. I am not changing anything. There was an interesting question on MO which OP removed by some ...
0
votes
0answers
19 views

What are the rank 3 boolean intervals [H,G], with G simple group?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$. The lattice $B_{3}$ is the following: Question: What are the rank $3$ boolean intervals of the form $[H,G]$, ...
3
votes
2answers
77 views

Does the collection of coverings on a set $X$ form a lattice when ordered by refinement?

This is a follow-up question to this question, prompted by a comment in Todd Trimble's answer. Let $X\neq \emptyset$ be a set. We say that $U\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a proper ...
1
vote
1answer
68 views

Order on the collection of coverings

Let $X\neq \emptyset$ be a set. We say that $U\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a proper covering if $\bigcup U = X$, and for $a\neq b\in U$ we have $a\not\subseteq b$. Let $\text{...
0
votes
0answers
69 views

Weak Topology and Domain Theory: Which topology on the function domain restricts to the weak topology on C([0,1])?

Let $\mathbb{IR}$ be the interval domain over the set $\mathbb{R}$ of real numbers, defined by: $$\mathbb{IR} := \{ [a,b] \mid a, b \in \mathbb{R}, a \leq b\} \cup \{ \mathbb{R}\},$$ and ordered by ...
4
votes
1answer
127 views

Dye's Theorem for real von Neumann algebras

Dye's classical theorem from 1955 states that if $\theta$ is a projection orthoisomorphism (i.e., a bijection between the projective lattices that preserves orthogonality - it then automatically ...
0
votes
0answers
55 views

Image of composition of integral upper triangular matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $\text{im}(AB)$ in terms of $\text{im}(A)$, $\text{im}(B)$, unions, intersections, determinants, ...
4
votes
0answers
59 views

Is there an interval of finite groups, at index n, with strictly more elements than the subgroup lattice of any group, of order n?

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $s(n):= max\{|\mathcal{L}(G)| \text{ for } |G|=n \}$. There is an OEIS page for the sequence $s(n)$: A018216 1, 2, 2, 5, 2, ...
1
vote
2answers
230 views

A good upper-bound for the cardinal of an interval of finite groups

This post is a relative version of General bound for the number of subgroups of a finite group Let $[H,G]$ be a interval of finite groups with $|G:H| = n$. Question: What is a good upper-bound of $|[...
0
votes
0answers
59 views

Addition Operations in Complete Lattices

Given a complete bounded lattice $L$, what do we know about the possibility of defining an addition operation $+$ that, broadly speaking, behaves like arithmetical addition? By this, I mean that as ...
2
votes
0answers
115 views

Nonvanishing of the dual Euler totient on boolean intervals of finite groups

The rank $n$ boolean lattice $B_n$, is the subset lattice of $\{1,2, \dotsm n \}$. Let $[H,G]$ be a boolean interval of finite groups. Its Euler totient is defined by $$\varphi(H,G):=\sum_{K \in ...
1
vote
2answers
60 views

What does the existence of self complemented elements tell us about a complete lattice?

Let $L$ be a complete lattice with an involution operation $*$ (a unary operation such that for any $x, y \in L$, $x \leq y$ implies $x^{*} \geq y^{*}$). Now, suppose that there is an element of $L$ ...
28
votes
1answer
736 views

What is the meaning of this analogy between lattices and topological spaces?

Let me add one more edit to help explain why this is a serious question. Theorem 5 below is a sort of lattice version of Urysohn's lemma, and it has essentially the same proof. Theorem 6, the famous ...
12
votes
1answer
533 views

Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way: Let $X$ be a set and let $\mathcal{P}...
2
votes
1answer
70 views

Topology with no direct lower neighbor

Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y] = \{p\in P: x\leq p \leq y\}$. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. The set $\text{Top}(X)$ is a complete ...
3
votes
1answer
84 views

“Discrete jumps” in the collection of all topologies on a set $X$

Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y] = \{p\in P: x\leq p \leq y\}$. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. The set $\text{Top}(X)$ is a complete ...
1
vote
1answer
89 views

Lower neighbors in the lattice of topologies

Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y) = \{p\in P: x\leq p < y\}$, and $(x,y]$ is defined in an analogous manner. For any set $X$, let $\text{Top}(X)$ denote the set of topologies ...
0
votes
1answer
101 views

Is the set of Cauchy spaces a lattice? [closed]

Is the set of all Cauchy spaces (ordered by set-theoretic inclusion) on some (fixed) set: a join-semilattice? a meet-semilattice? a complete lattice?
1
vote
0answers
72 views

An optimal lower bound related to generators in a boolean interval of finite groups

Let $[H,G]$ be a rank $n$ boolean interval of finite groups (i.e. $[H,G] \simeq B_n$ as lattice). Let the set $E = \{ g \in G \ | \ \langle H,g \rangle = G \}$ Remark: If $g \in E$ then $Hg \...
2
votes
1answer
158 views

Pseudocomplements in the lattice of topologies

Given a set $X\neq \emptyset$ it is well-known that the collection $\text{Top}(X)$ of all topologies on $X$ is a (complete) lattice with respect to $\subseteq$. Let $0$ denote the smallest element of ...
4
votes
0answers
68 views

Cases of equality in Daykin's theorem

Let $A$ and $B$ be sets of subsets of $\{1, \dots, n\}$, and let $A \wedge B = \{a \cap b : a\in A, b\in B\}$, $A\vee B=\{a\cup b: a\in A, b\in B\}$. Then $$ |A \wedge B| |A\vee B| \geq |A||B|, $$ as ...
6
votes
1answer
566 views

Positivity of the alternating sum of indices for boolean interval of finite groups

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice. Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$. Let the alternative sum ...
1
vote
1answer
103 views

Can the reversed lattice of a subgroups interval be represented?

Let $G$ be a finite group and $H$ a subgroup. The interval $[H,G]$ is the lattice of overgroups of $H$. It is an open problem to know if every finite lattice can be represented by such an interval (...
9
votes
1answer
172 views

Are these two quotients of $\omega^\omega$ isomorphic?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say $f\simeq_{\text{fin}} g$ if there is $n\in \omega$ such that $f(k) = g(k)$ for all $k\geq n$. ...
15
votes
4answers
465 views

Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
0
votes
1answer
29 views

Characterization of Dedekind complete Riesz spaces by strictly positive functionals

I was browsing throughout the literature and I found the following fact: Every $\sigma$-Dedekind complete Riesz space $E$ which admits a strictly positive functional $f$ is Dedekind complete. I ...
5
votes
2answers
250 views

A weak kind of fixed point

Let $X$ be a set and let $\cal A$ be a non-empty subset of $P(X)$ with the property that whenever $A_1 \subseteq A_2 \subseteq \cdots $ is an increasing chain of elements of $\cal A$ then $\cup_i A_i \...
1
vote
0answers
100 views

The lattice of graphs under vertex abstractions

I am curious to know if the following structure has been studied, or if anything similar is in the literature. For $n \in \mathbb{N}$, let $G = ([n],E)$ be a digraph. A partition of a subset $V$ of $[...
3
votes
2answers
98 views

About Kurosch-Ore theorem

Where can I find the proof of Kurosch-Ore theorem in lattice theory? The statement of this theorem is: Let $L$ be a modular lattice with $0$ and $1$ that satisfies both chain conditions. Then for any ...
-1
votes
1answer
169 views

Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?

Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and $...
3
votes
0answers
90 views

What are universal abstract $\sigma$-algebras on $\sigma$-frames?

Originally asked on MSE. In this paper, the authors make the following definitions: An (abstract) $\sigma$-algebra is a boolean algebra with countable joins. A $\sigma$-frame is a bounded lattice ...
1
vote
1answer
174 views

“Zorn's Lemma guarantees that all algebraic frames are spatial.” Why?

In at least two papers (here and here) Jorge Martínez and Eric R. Zenk say that Zorn's Lemma implies that all algebraic frames are spatial. However, I haven't been able to find an actual explanation ...
2
votes
1answer
77 views

Is this a sufficient condition for distributivity of a lattice?

I asked this question here on Math.SE but uptil now it was not answered. So I decided to give it a try here. Thank you in advance. If a lattice $L$ is distributive then it can be shown that for $a,b,...
3
votes
1answer
154 views

Can a length n distributive lattice be embedded into Bn?

Let $\mathcal{L}$ be a finite distributive lattice, then it is known that it can be embedded into a finite boolean lattice (see theorem 8.5. p91 in this note). Let $n$ be the length of $\mathcal{...
1
vote
0answers
102 views

A finite distributive lattice which may be represented as the normal subgroup lattice of a supersolvable group

Is there a supersolvable group $G$ with the lattice of all its normal subgroups, order-isommorphic to the 18-element lattice of down-sets of this poset: ? It has been proved that not every finite ...
0
votes
1answer
80 views

Prime ideals containing the finite members of ${\cal P}(\omega)$

Let ${\frak P}$ denote the collection of prime ideals containing the finite members of ${\cal P}(\omega)$, and order ${\frak P}$ by set inclusion. What is the cardinality of ${\frak P}$, and what's ...
2
votes
3answers
126 views

Non-principal prime ideals in infinite distributive lattices

Given an infinite distributive lattice $L$, does $L$ contain a non-principal prime ideal $I$, or a non-principal prime filter $F$? ($I$ is said to be principal if there is $x\in L$ such that $I=\{y\in ...
5
votes
1answer
114 views

An equivalent definition for the adiamond lattices

A lattice is called adiamond if it admits no sublattice equivalent to the diamond lattice $M_3$ below: The top interval of a lattice is the interval between the meet of all the maximal elements and ...
2
votes
2answers
108 views

Are lattices quotients of their Dedekind-MacNeille completion?

Let $L$ be a lattice and let $\textbf{DM}(\cdot)$ denote the Dedekind-MacNeille completion. Is there a lattice $L$ that is not a quotient of $\textbf{DM}(L)$? And what if we generalise this question ...
32
votes
3answers
616 views

How is the free modular lattice on 3 generators related to 8-dimensional space?

Here are three facts which sound potentially related. What are the actual relationships? In 1900, Dedekind constructed the free modular lattice on 3 generators as a sublattice of the lattice of ...
9
votes
0answers
184 views

Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion of finite groups. Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ ,...
1
vote
0answers
110 views

Examples of value quantales

In his paper "Quantales and continuity spaces" R. C. Flagg gives the following examples of value quantales: the lattice $\bf{2}$ of truth values with usual addition, the lattice $\mathbb{R}_{+}$ of ...
0
votes
0answers
92 views

Properties of a specific antichain of a lattice formed by the cartesian product of finite ordered sets

Introduction Let $X$ be a poset of all $n$-tuples, $x = (x_1, x_2, ..., x_n)$, where $0 \leq x_i \leq m_i - 1$ for $i = 1, ..., n$ together with the relation $x \prec y$ defined so that for $y=(y_1,...
2
votes
0answers
73 views

Lattice Flatness Measure

I am looking for the definition of a flatness measure in lattice theory. More generally, I am looking at finite-height lattices and I want to measure their complexity, with a perfectly flat lattice ...
1
vote
0answers
59 views

Standard name / symbol for intersection in Brouwerian lattices

A Brouwerian lattice has a lower adjoint $\cdot - B$ to $B\lor\cdot$. It is called pseudodifference. The main reference is http://www.jstor.org/stable/1969038 Once you have pseudodifference, you can ...
1
vote
0answers
67 views

Is there an official name for the intersection of the join-irreducible representations of two lattice elements?

Given a lattice provided with a join-irreducible representation of its elements, there is a natural "intersection" operator $A \mathbin{\dot\cap} B$ that returns the join of the setwise intersection ...
1
vote
0answers
83 views

Relationship of ${\cal P}(\omega)/fin$ and ${\cal L}$

Define ${\cal L}$ as in this question: the set of functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ at at ...
3
votes
1answer
156 views

Properties of the interval topology of the lattice of functions

Let $(P,\leq)$ be a poset. 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\...
1
vote
1answer
80 views

Is there a surjective lattice homomorphism $f: {\cal L}\to \mathbb{N}^\mathbb{N}$?

Let ${\cal L}$ be defined as in this question. Is there a surjective lattice homomorphism $f: {\cal L}\to \mathbb{N}^\mathbb{N}$, where $\mathbb{N}^\mathbb{N}$ is the set of all functions, ordered ...
11
votes
1answer
486 views

How big is the lattice of all functions?

Define the lattice $(\mathcal{L},\prec)$ as the set of all function $f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ ...