All Questions
Tagged with lattice-orders order-theory
538
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Galois connections give rise to complete lattices
I am reading Introduction to Lattices and Order, second edition, by Davey and Priestly. On page 161, it says
Every Galois connection $(^\rhd,^\lhd)$ gives rise to a pair of closure operators, $^{\rhd\...
- 1,889
-2
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23
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The supermodularity of probability of intersection [closed]
Given a finite sample space $E$, let $E=\{A_1,A_2,\dots,A_n\}$ be a collection of random events.Then, is $f(S)=\mathbb{P}\{\cap_{A_i\in S}A_i\}$ a supermodular function for $S\subseteq E$?
- 31
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If sub-universe $S$ of lattice has congruence $\theta$, does the lattice have a congruence $\lambda = \theta \cap S^2$? [duplicate]
Let $(L, \lor , \land )$ be a lattice and $S$ a sub-universe of the lattice. A sub-universe of a lattice will be any subset of the lattice set that is non-empty and closed under $\land$ and $\lor$. ...
- 3,468
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40
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Understanding the definition of congruences over a lattice
Let $(L, \land, \lor)$ a lattice and $\theta$ a binary relation over $L$. We say $\theta$ is a congruence iff
$$
x_0\theta x_1, y_0 \theta y_1 \Rightarrow (x_0 \lor y_0)\theta(x_1 \lor y_1)
$$
(and ...
- 3,468
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1
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45
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Number of lattices over a finite set
I'm interested in finding a general formula for the number of lattices possible over a finite set $S$ as a function of the set's cardinality.
For instance, how many lattices over $\{1, 2, 3\}$ are ...
- 3,468
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52
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Extending the $M_3,N_5$ theorem from distributive lattices to frames
It is known that a lattice $L$ is distributive if and only if it does not contain the diamond $M_3$ or the pentagon $N_5$ as sublattices.
A complete lattice is one in which every subset has an infimum ...
- 53
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Congruences on the pentagon lattice $\mathcal{N}_5$
Let $\mathcal{N}_5$ refer to the Pentagon lattice, or the lattice generated by the set $\{0, a, b, c, 1\}$ subject to $1 > a$, $1 > c$, $a > b$, $b > 0$ and $c > 0$.
My aim is to find ...
- 497
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1
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52
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Lattice with supermodular height function is lower semimodular
Question
Let $(L,\leq)$ be a lattice of finite length and let its height function $h$ be supermodular, meaning that
$$h(x \wedge y) + h(x \vee y) \geq h(x) + h(y) \quad \forall x,y\in L.$$
Does it ...
- 23.1k
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How to get the distributive law for an l-group?
In Birkhoff an l-group G is defined as a group that is also a poset and in which group translation is isotone:
\begin{gather*}
x\leq y\implies a+x+b\leq a+y+b\;\forall a,x,y,b\in G,
\end{gather*}
and ...
- 23
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How to get the height function for modular lattices?
In these notes, it is said that for modular lattices of finite lengths the height function
\begin{gather*}
h(x)=lub\{l(C):C=\{x_0,...,x_n:x_0=O\prec...\prec x_n=x\}\}
\end{gather*}
obeys
\begin{gather*...
- 33
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1
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51
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Alternative characterization of distributive lattice
Let $(X,{\leq},{\wedge},{\vee})$ be a lattice.
The lattice is called distributive if for all $x,y,z\in X$ both distributive laws hold:
$$
x \wedge (y \vee z) = (x \wedge z) \vee (y \wedge z)
\quad\...
- 23.1k
2
votes
1
answer
53
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Möbius function of distributive lattice only takes values $\pm 1$ and $0$.
In this Wikipedia article, I found the statement
[...] shares some properties with distributive lattices: for example, its Möbius function takes on only values 0, 1, −1.
My question is: How it can ...
- 23.1k
2
votes
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41
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The join of two set partitions in the refinement order
Let $X$ be a set. The set $\Pi_X$ of all partitions of $X$ is partially ordered via the refinement order, which is defined by $\alpha \leq \beta$ if and only if for each block $A\in \alpha$ there is a ...
- 23.1k
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Reconstructing a closure operator from a set of fixed points
Let $L$ be a lattice, not necessarily complete. We define a closure operator as a function $f\colon L\to L$ which is:
idempotent, $f(f(x)) = f(x)$,
isotone, $x\leq y \Rightarrow f(x) \leq f(y)$,
...
- 386
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72
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Zorn's lemma: counterexample to chain with upper bound?
The premise required for invoking Zorn's lemma is that every chain in $X$ have an upper bound. So that makes me wonder: what is a good example of a poset $X$ for which that property is false? That is ...
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