# Tagged Questions

**0**

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### approximate coordinates in a one dimensional lattice

suppose I have a finite set of real numbers ${r_1, \ldots r_n \in \mathbf{R} }$ and a single real number $x \in \mathbf{R}$. Is there a fast algorithm for finding integer numbers ${i_1, \ldots i_n \in ...

**4**

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**1**answer

326 views

### Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...

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**2**answers

121 views

### Are there atomistic ortholattices which are not modular?

Let $L$ be an atomic ortholattice. We say that two elements $a$ and $b$ of $L$ are orthogonal if $a\leq b^\perp$. If $L$ is orthomodular then every element of $L$ can be written as a join of pairwise ...

**4**

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**2**answers

374 views

### Product-Decomposition of distributive lattices

EDIT
I now (strongly) believe that the following claim answers my question (see the text below). However, if it does, then I am sure that it is known. It is not difficult to prove and the question ...

**2**

votes

**1**answer

278 views

### Name of a lattice-property

Assume that we have a complete lattice $(L,\leq)$.
I would like to know whether the following property has a specific name and whether lattices with this property have been studied somewhere:
For ...

**1**

vote

**1**answer

301 views

### Complete De Morgan algebra

Recall that an algebra $(A,\sim)$ is a De Morgan algebra if $A$ is a bounded distributive lattice and $\sim$ is a unary operation which satisfies:
${\sim} (x\vee y)={\sim} x\wedge {\sim} y$ and ...

**2**

votes

**1**answer

328 views

### Extending a complete lattice to get a “nice” Boolean lattice

Suppose we have a complete lattice. Which additional axioms (e.g. distributivity axioms) are needed to obtain a Boolean lattice in which complement(a) = lub{b | b /\ a = bottom} = glb{b | b / a = ...

**15**

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**1**answer

673 views

### Codes, lattices, vertex operator algebras

At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:
Finally, we cannot resist calling attention ...

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132 views

### Bogus(?) proof for equivalence of normal and dual distributivity conditions in lattices

I am attempting to prove the equivalence of the following two definitions of distributive lattices:
$(a \lor b) \land c = (a \land c) \lor (b \land c)$
$(a \land b) \lor c = (a \lor c) \land (b \lor ...

**4**

votes

**1**answer

570 views

### Can the Knaster-Tarski theorem be proved using the Schroeder-Bernstein theorem?

The reverse can be done easily and the proof is well known I am wondering if the exact same argument can be used to prove reverse as well.

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vote

**1**answer

380 views

### A complete lattice of functions

Let $D$ be a set, $\mathbb{N_0}$ the set of natural numbers including zero. Let $P$ be the set of all functions from $D$ to $\mathbb{N_0}$, i.e. $P = \lbrace m \mid m: D \rightarrow \mathbb{N_0} ...

**14**

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**3**answers

2k views

### Proving that a poset is a lattice

I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably ...

**21**

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**4**answers

3k views

### Gossip about Grothendieck and distributive lattices

In Gian-Carlo Rota's Indiscrete Thoughts, there a list of mathematical gossip among which one reads:
[...] What would have happened [...] if Grothendieck had known the theory of distributive ...

**2**

votes

**3**answers

613 views

### Lattice of subcategories: subobject classifier in Cat

Two short questions:
Is there any work classifying the lattice of subcategories of an arbitrary (sufficiently small) category $\mathcal{C}$, similar to the way that the set of subsets of set ...

**0**

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**0**answers

162 views

### Vector-valued valuations on lattices

There's a fair amount of work on valuations on (modular) lattices, by which I mean functions $v : \mathcal{L} \rightarrow R$ that satisfy the modular expression
$$v(x) + v(y) = v(x \wedge y) + v(x ...

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vote

**2**answers

917 views

### Product lattice

Could someone explain how to construct a product lattice, or point me to an explanation on the web?

**3**

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**1**answer

888 views

### LUB GLB on a Lexicographically ordered complete lattice product

I am trying to define the LUB and GLB on a product of lattices that are partially ordered lexicographically.
Is there any papers or help that I could read up on? I would particularly like proofs on ...

**5**

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**3**answers

1k views

### Constructing a metric over a lattice

Consider a lattice $({\cal L}, \wedge, \vee)$ with an antimonotonic function $f: {\cal L} \rightarrow {\mathbb R}$ defined on it (i.e $x \preceq y \implies f(x) \ge f(y)$).
$f$ is said to be ...

**25**

votes

**6**answers

5k views

### Good lattice theory books?

A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - ...