Timeline for Set of ideals of the set of finite subsets of $\mathbb{N}$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 2, 2015 at 6:58 | comment | added | Dominic van der Zypen | Very nice @JosephVanName - thanks for pointing out this approach! | |
| Jun 1, 2015 at 17:12 | comment | added | Joseph Van Name | If $X$ is a join-semilattice, then $\mathcal{I}(X)$ is an algebraic lattice, and if $L$ is an algebraic lattice, then the set $K(L)$ of compact elements in $L$ is a join-semilattice. If $L$ is an algebraic lattice, then $L\simeq\mathcal{I}(K(L))$, and if $X$ is a join-semilattice, then $X\simeq K(\mathcal{I}(X))$. The lattice $P(\mathbb{N})$ is an algebraic lattice and $K(P(\mathbb{N}))=F(\mathbb{N})$. Therefore, $\mathcal{I}(F(\mathbb{N}))=I(K(P(\mathbb{N})))\simeq P(\mathbb{N})$. See newton.case.edu/papers/algfca.pdf for a proof of this duality. | |
| Jun 1, 2015 at 17:11 | comment | added | Joseph Van Name | Good answer. A more high-level way to see this isomorphism is through the duality between join-semilattices and algebraic lattices. Algebraic lattices are described in detail in the book A Course in Universal Algebra by Burris and Sankappanavar. Of course, the duality between join-semilattices and algebraic lattices is a part of a larger framework of dualities between ordered sets and complete lattices. | |
| Jun 1, 2015 at 9:26 | vote | accept | CommunityBot | ||
| Sep 2, 2015 at 16:36 | |||||
| Jun 1, 2015 at 9:20 | history | answered | Dominic van der Zypen | CC BY-SA 3.0 |