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Results tagged with lattice-theory
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user 75
The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.
3
votes
Accepted
Lattice homomorphism from ${\cal Id}(L)$ onto $L$
Here's a counterexample. Let $L=\{0,1,x_0,x_1,x_2,\dots,y\}$, where $x_0<x_1<x_2<\dots$ and $y$ is incomparable with every $x_n$. Then the only non-principal ideal in $L$ is $I=\{0,x_0,x_1,\dots\}$; …
- 31.2k
8
votes
Accepted
Cardinality of pairwise non-isomorphic complete lattices on an infinite cardinal $\kappa$
There is a trivial upper bound of $2^\kappa$. In fact, this bound can already be achieved for complete totally ordered sets. For instance, given any subset $A\subseteq\kappa+1$, let $L_A$ be obtaine …
- 31.2k
8
votes
Accepted
Does ${\cal Id}(L) \cong {\cal Id}(K)$ imply $L\cong K$?
No; in fact, we can canonically recover $L$ from $\mathcal{Id}(L)$ as the sublattice of compact elements (that is, elements $x$ such that whenever $x=\bigvee S$, there is a finite subset $F\subseteq S …
- 31.2k
4
votes
Complete non-isomorphic lattices with injective complete homomorphisms between them?
For a simple example with complete total orders, take $L=\{0\}\cup[1,2]$ and $K=\{-1,0\}\cup[1,2]$.
- 31.2k
11
votes
Accepted
Embedding finite lattices into the lattice of partitions of a finite set
Yes, this is apparently a fairly hard theorem of Pudlak and Tuma (or at least I assume it is hard, because it seems to have been an open problem for decades before they finally proved it in 1980).
- 31.2k