**0**

votes

**0**answers

46 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, ...

**0**

votes

**0**answers

79 views

### Dirac functional embedding [on hold]

I got the following set up:
Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with the pointwise ...

**4**

votes

**0**answers

57 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

**2**answers

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

**0**answers

53 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

**0**answers

114 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

**2**answers

57 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

**1**answer

729 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

**1**answer

529 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

**1**answer

69 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

**1**answer

81 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

**1**answer

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

**1**answer

97 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

**0**answers

71 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

**1**answer

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

**0**answers

67 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

**1**answer

563 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

**1**answer

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

**1**answer

171 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

**4**answers

453 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

**0**answers

22 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

**2**answers

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

**0**answers

99 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

**2**answers

97 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

**1**answer

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

**0**answers

89 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

**1**answer

173 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

**1**answer

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

**1**answer

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

**0**answers

101 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

**1**answer

79 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

**3**answers

125 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

**1**answer

113 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

**2**answers

105 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

**3**answers

613 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

**0**answers

182 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

**0**answers

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

**0**answers

90 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

**0**answers

72 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

**0**answers

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

**0**answers

66 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

**0**answers

82 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

**1**answer

153 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

**1**answer

79 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

**1**answer

485 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 ...

**-1**

votes

**2**answers

133 views

### Complete non-isomorphic lattices with injective complete homomorphisms between them?

Are there complete lattices $L, K$ such that
$L\not\cong K$;
there are injective complete lattice homomorphisms $i:L\to K$ and $j: K\to L$
?

**0**

votes

**1**answer

84 views

### Does ${\cal Id}(L) \cong {\cal Id}(K)$ imply $L\cong K$?

For any lattice $L$ we denote the complete lattice of the ideals of $L$ by ${\cal Id}(L)$. Are there non-isomorphic lattices $L\not \cong K$ such that ${\cal Id}(L) \cong {\cal Id}(K)$?

**0**

votes

**1**answer

116 views

### Cardinality of pairwise non-isomorphic complete lattices on an infinite cardinal $\kappa$

Suppose $\kappa$ is an infinite cardinal. Let $\cal S$ be a collection of pairwise non-isomorphic complete lattices on the ground set $\kappa$. What cardinality can $\cal S$ have at most? Is the ...

**3**

votes

**0**answers

73 views

### Hausdorff spaces with lattice isomorphism between the topologies [closed]

For $i=1,2$ let $(X_i,\tau_i)$ be Hausdorff spaces such that the lattices $\tau_1, \tau_2$ are isomorphic.
Does this imply that $(X_1, \tau_1) \cong (X_2,\tau_2)$?
(This is a follow-up question to $...

**0**

votes

**1**answer

54 views

### $T_2$-spaces such that the lattices of open sets can be embedded into each other

Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$.
Does this imply that $(X,\tau)\cong (Y,\sigma)$?