**0**

votes

**0**answers

54 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 ...

**2**

votes

**0**answers

45 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

50 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 ...

**0**

votes

**0**answers

24 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

68 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

132 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: ...

**1**

vote

**1**answer

70 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 ...

**13**

votes

**1**answer

441 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

118 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

77 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

112 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

62 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

46 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 $(X,\tau)\cong (Y,\sigma)$?

**0**

votes

**1**answer

20 views

### Reducing join-incomplete lattice homomorphisms to homomorphisms with co-domain ${\bf 2}$

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) > \bigvee_L f(S).$
Is it true ...

**0**

votes

**0**answers

62 views

### An equality for the trace of the join of a non-degenerate indecomposable system of projections in a finite factor

Let $M \subset B(H)$ be a finite factor (see for example here p2, or there) with a trace $tr$.
The subset of projections of $M$ is naturally a lattice, noted $(\mathcal{P}(M), \wedge, \vee)$.
A ...

**2**

votes

**1**answer

130 views

### Is the top interval of a finite distributive lattice an hypercube lattice?

Let $(L,\wedge,\vee)$ be a finite distributive lattice. Let $M$ be the (unique) maximum element. An element $a \in L$ is called maximal if $a \le a' < M$ implies $a = a'$. Let $b = ...

**0**

votes

**0**answers

28 views

### Characterization of complete lattices with join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) > \bigvee_L f(S).$
How can ...

**0**

votes

**1**answer

59 views

### Contracting join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_L f(S).$
If $f:L\to ...

**1**

vote

**1**answer

52 views

### Simplyfing join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_L f(S).$
Is the ...

**0**

votes

**1**answer

101 views

### Lattices without prime ideals

If $\kappa$ is an infinite cardinal, is there a lattice $L$ of cardinality $\kappa$ such that $L$ contains no prime ideals?

**-1**

votes

**1**answer

54 views

### Lattice homomorphism from ${\cal Id}(L)$ onto $L$

For any lattice $L$ we denote the complete lattice of the ideals of $L$ by ${\cal Id}(L)$. If $L$ is complete, is there a lattice homomorphism from ${\cal Id}(L)$ onto $L$?

**0**

votes

**2**answers

111 views

### Join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_L f(S).$
Suppose ...

**3**

votes

**1**answer

86 views

### Incomplete lattice homomorphisms between complete lattices (2)

Let $L, K$ be complete lattices. A lattice homomorphism $f: L\to K$ is said to be incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_K f(S).$
Consider the ...

**0**

votes

**1**answer

100 views

### Incomplete lattice homomorphisms between complete lattices

Let $L, K$ be complete lattices. A lattice homomorphism $f: L\to K$ is said to be incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) \neq \bigvee_K f(S).$
Suppose that ...

**0**

votes

**2**answers

121 views

### Embedding finite lattices into the lattice of partitions of a finite set

For any set $X$ we denote by $\text{Part}(X)$ the set of all partitions of $X$, ordered by the refinement ordering. It is well known that this is a complete lattice for all sets $X$.
Let $L$ be a ...

**4**

votes

**1**answer

150 views

### 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 ...

**0**

votes

**1**answer

64 views

### 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 ...

**0**

votes

**1**answer

75 views

### 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 ...

**4**

votes

**1**answer

121 views

### 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, ...

**5**

votes

**1**answer

352 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 ...

**4**

votes

**0**answers

315 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 ...

**3**

votes

**0**answers

128 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.)
...

**4**

votes

**2**answers

224 views

### 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 ...

**-3**

votes

**2**answers

86 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 ...

**2**

votes

**0**answers

117 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 ...

**2**

votes

**0**answers

79 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 ...

**5**

votes

**1**answer

110 views

### 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 ...

**-1**

votes

**3**answers

115 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 $f:L\to\omega^\omega$?

**2**

votes

**1**answer

62 views

### 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 ...

**1**

vote

**1**answer

78 views

### 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 ...

**0**

votes

**2**answers

107 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 ...

**7**

votes

**0**answers

140 views

### 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) ...

**2**

votes

**0**answers

47 views

### 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 ...

**1**

vote

**0**answers

85 views

### 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 ...

**9**

votes

**2**answers

464 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 ...

**0**

votes

**0**answers

48 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 ...

**1**

vote

**2**answers

101 views

### 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?

**-1**

votes

**1**answer

180 views

### 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 ...

**8**

votes

**0**answers

210 views

### 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 $P(\omega)$ ...

**0**

votes

**1**answer

63 views

### 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 ...