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### The groups with symmetric subgroups lattice

Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice.
If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the ...

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### Set of ideals of the set of finite subsets of $\mathbb{N}$

In any lattice $L$, an ideal is a subset $I\subseteq L$ that is downward closed, and moreover $a,b\in I$ implies $a\vee b\in I$. We denote by ${\cal I}(L)$ the set of ideals of $L$, ordered by set ...

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### Dedekind-MacNeille completion of the strictly increasing members of $\omega^\omega$

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ ordered by $f\leq g$ iff $f(n) \leq g(n)$ for all $n\in \omega$.
Set $K = \{f\in \omega^\omega: m<n\in \omega \implies ...

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### Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every $x′>x$ and $y′>y$, we have
\begin{equation*}
f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′).
\end{equation*}
Suppose $f$ and $g$ are supermodular, ...

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317 views

### Generalization of the fundamental theorem of cyclic groups

Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ is cyclic iff it admits no two different subgroups with the same order.
proof: see ...

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278 views

### Order theory as a foundation of mathematics?

I know the followings kinds of formalization of mathematics:
based on set theory (e.g. ZFC)
based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example)
based on category ...

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123 views

### A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height

Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.)
...

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### Counterexample on completely distributive lattices

I would like to see an example of a complete lattice $C$ which is both a frame and a dual-frame, i.e. finite meets distribute over arbitrary joins and finite joins distribute over arbitrary meets ...

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76 views

### does a lattice have a minimal item [closed]

A lattice (L, ≤) is a partially ordered set, where each two elements of the set have a least upper bound and a greatest lower bound. Let's consider the situation where L is finite.
I think that ...

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110 views

### Are all monomorphisms in the category of bounded lattices regular?

Let $\mathbf{Lat}_{01}$ be the category of bounded lattices with lattice homomorphisms that respect the smallest and the largest element. Is there a monomorphism in $\mathbf{Lat}_{01}$ that is not ...

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75 views

### About some distributive laws in the Bousfield lattice

It is know that for any $\alpha$-well generated tensor triangulated category $\mathcal{T}$ the collection of Bousfield classes forms a set, furthermore this set is a complete lattice, denoted by ...

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### How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of G. D. Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and ...

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107 views

### Finite distributive lattices not contained in $\omega^\omega$

If we consider $\omega^\omega$ as a lattice with component-wise join and meet, is there a finite distributive lattice $L$ so that there is no injective lattice homomorphism $f:L\to\omega^\omega$?

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### Is $[0,1]^\kappa$ an affine complete lattice?

A $k$-ary function $f$ on a bounded distributive lattice
$L$ is called compatible if for any congruence relation $\theta$ on $L$ and $(a_i, b_i)\in \theta$ for $i=1,\ldots,k$ we always have ...

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### What is the smallest partition lattice PART(M) containing the lattice P(N) of subsets of a finite set of N elements

It's been known for 35 years that every finite lattice can be embedded in a finite partition lattice (Pudlak and Tuma, Algebra Universalis 1980, Volume 10, Issue 1, pp. 74--95).
I don't follow the ...

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101 views

### Continuous image relation on topological spaces

Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by
for $X, Y \in ...

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### Decidabilty of the Hilbert lattice and quantum logic

What is known about the decidability of (first-order formulas in) the structure $(\mathcal{L}(H),\leq)$, where $\mathcal{L}(H)$ is the collection of all closed linear subspaces of a (separable) ...

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### Lattice of subobjects of a particular coproduct

I have the following situation: $\mathcal C$ is a (good enough, say Grothendieck) Abelian category and $F:\mathcal C\to \mathcal C$ is self-equivalence. Given an object $C$ in $\mathcal C$, what can I ...

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### Lattice: Join of orthogonal elements is set union?

I am studying a lattice of closed sets generated by an orthogonality relation $\perp$ with set inclusion as ordering relation and meet and join defined as $(A \land B)=A \cap B$ and $(A \lor B) = (A ...

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460 views

### Can any finite lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.
Question: Can any finite lattice be realized as ...

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47 views

### Is the core of an atom in lattice of group topologies a coatom?

Let $(G,\mathcal T)$ be an abelian topological group such that for any nontrivial group topology $\mathcal S$ on $G$ with $\mathcal S\subseteq \mathcal T$ we have $\mathcal S = \mathcal T$.
Let ...

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### Order-preserving image of a complete lattice

If $L$ is a complete lattice and $P$ is a poset and $f: L\to P$ is an order preserving surjective map, does this imply that $P$ is a (complete) lattice?

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### How to recognize if a lattice is distributive? [closed]

I know that a Boolean lattice must be distributive.
But what with these lattices? Are these distributive?
$\hskip0.7in$
How to recognize which lattices are distributive or not only by looking on ...

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### Is there a notion analogous to separability but requiring definable rather than countable sets?

Among models of $\lambda$-calculus, some like the Bohm tree model have the property that every element is a directed sup of definable elements, whereas others like the $D_\infty$ and $P(\omega)$ ...

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### Lower bounds on the rank of a unimodular lattice, given the binlinear pairing of a subset of basis vectors

I have some questions on lower bounds on the rank of unimodular lattices given the bilinear pairing of a subset of its basis is known.
Let $\Lambda$ be an odd, unimodular matrix of signature ...

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### For any two noncrossing partitions $p, q$ of $n$, is the graph of geodesics from $p$ to $q$ in $NC(n)$ connected?

Let $NC(n)$ denote the lattice of noncrossing partitions of $n$, and let $G$ denote the Hasse diagram of $NC(n)$ with respect to covering relations, viewed as an undirected graph.
I'm interested in ...

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### When is the homomorphism poset between posets a lattice?

Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. There is a natural ordering relation on $\text{Hom}(P,Q)$ given by $f\leq ...

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### Unique representability of bounded distributive lattices

Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space.
A poset $(P,\leq)$ is called ...

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### Minor ordering for finite graphs

Let $\mathcal{G}_{<\omega}$ be the set of graphs $G = (V,E)$ such that $V = \{0,\ldots,n\}$ for some $n \geq 0$ and $E \subseteq \mathcal{P}_2(V) = \{\{a,b\} : a,b \in V \textrm{ and } a\neq b\}$. ...

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### What properties of lattice are preserved in a weak lattice structure [closed]

A lattice structure requires that every two elements have a join and a meet.
Suppose we consider instead posets in which for every two elements $x,y$, if there exists an element greater than both of ...

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### About a construction of Borel $\sigma$-algebra associated to a lattice

Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$).
Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ ...

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### Uniformizing a relation on ordered sets

Suppose $A$ and $B$ are (complete) ordered sets. Suppose $R\subseteq A\times B$, and
$f(a)=\inf\{b : (a,b)\in R\}$
$g(b)=\inf\{a : (a,b)\in R\}$
then what can we call $f$ and $g$? Perhaps there is ...

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### Normal subgroup lattice of the group $U_{6n}$

I need to find the normal subgroup lattice of the group $U_{6n} = \langle a,b | a^{2n} = b^ 3= 1, a^{-1}ba = b^{-1}\rangle$. To the best of my knowledge this group was introduced at first by GORDON ...

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### When does an orthomodular projection lattice have a non-trivial centre?

When does an orthomodular lattice $L$ of projections onto a given Hilbert space have a non-trivial centre $Z(L)$ and what can we generally say about the cardinality of $Z(L)$?

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### Relative Compactness vs Way Below in Locally Compact Hausdorff Spaces

Let $Y$ be a subset of a locally compact Hausdorff topological space $X$ and consider the following properties.
$\overline{Y}$ is compact.
Every open cover of $X$ has a finite subcover of $Y$.
...

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195 views

### Cancellative semigroup on a distributive lattice

Let $(S,\le)$ be a distributive lattice. Is there a semigroup structure on $S$ such that $S$ is cancellative and always $(x\wedge y)(x\vee y)=xy$?

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### Which Hyperspace Topologies Yield Topological Lattices?

At least on a continuum, the binary operations of intersection and union are Vietoris-continuous. But the Vietoris topology only applies the the collection of NONEMPTY closed subsets, and this means ...

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### Distributivity of group topologies on $\Bbb Z$

Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$.
It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice [1].
Is $(\mathcal L,\subseteq)$ distributive?
$$~$$
[1] ...

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430 views

### A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{trivial}, \mathcal T_{discrete}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$\mathcal ...

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### Self-duality of the subgroup lattice of $G\times H$

Let $G$ and $H$ be finite groups and $\gcd(|G|,|H|)=1$.
Suppose the lattice of all subgroups of $G\times H$ is self-dual. Is the lattice of all subgroups of $G$ self-dual?

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### A twisted self-dual subgroup lattice

A lattice $(X,\le)$ is twisted self-dual iff it is self-dual but there is not any self-duality $f:X\to X$ with $f\circ f=1_X$.
Is there any group with lattice of all its subgroups twisted self-dual?

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### Self-duality in a lattice

Is there any self-dual lattice $(X,\le)$ such that there is not any self-duality $f:X\to X$ such that $f\circ f = 1_X$?

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### Is it true that the set of points minimizing their distance to a multiset of intervals from a distributive lattice is an interval?

Let $(E, \preceq)$ be a finite distributive lattice, $H_E$ be the Hasse diagram of $E$ and $d$ be the distance on $E \times E$ defined as the length of the shortest path in $H_E$ between any pair of ...

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### Status of Barany's conjecture?

One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks:
Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$?
A ...

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### a characterization for cyclic groups [duplicate]

Let $G$ be a group of order $n$ and its subgroup lattice be order-isomorphic to that of $\Bbb Z_n$. Is $G$ cyclic?

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### generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?

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### Which complete lattices arise as images of the Galois connections induced by binary relations?

Any binary relation $R\subseteq X\times Y$ gives rise to a Galois connection between the powersets of $X$ and $Y$ in a well known way (on MO you can see it e. g. in this answer; in fact, such Galois ...

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### Do any Stone-like dualities have some self-dualities hidden inside them?

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...

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### $\mathcal{L}(H_i \subset G_i)$ distributive $\Rightarrow$ $\mathcal{L}(H_1 \times H_2 \subset G_1 \times G_2)$ modular?

Let $\mathcal{L}( G)$ be the lattice of subgroups of $G$ and $\mathcal{L}(H \subset G)$ the lattice of intermediate subgroups.
Definitions: A lattice $(L, \wedge, \vee)$ is distributive if, ...

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### Products of maximal inclusions of finite groups with a non-obvious intermediate

Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups.
Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...