Difference between revisions of "Funcoid bases/Basic results"
(Created page with "That $S$ is an upper set follows from every three supposedly equivalent conditions of the main conjecture. Thus we can assume that $S$ is an upper set while proving the conjec...") 
(No difference)

Latest revision as of 20:25, 13 April 2017
That $S$ is an upper set follows from every three supposedly equivalent conditions of the main conjecture. Thus we can assume that $S$ is an upper set while proving the conjecture.
We will denote $S' = S \cap \Gamma$ (where $\Gamma$ is the boolean lattice defined in "Funcoids are filters" chapter of the book).
Proposition (Under condition that $S$ is an upper set) we have $\bigwedge^{\mathsf{FCD}} S' = \bigwedge^{\mathsf{FCD}} S$.
Proof $\bigwedge^{\mathsf{FCD}} S' \sqsupseteq \bigwedge^{\mathsf{FCD}} S$ is obvious.
$\bigwedge^{\mathsf{FCD}} S = \bigwedge^{\mathsf{FCD}}_{K \in S} \bigwedge^{\mathsf{FCD}} \operatorname{up}^{\Gamma} K \sqsupseteq \bigwedge^{\mathsf{FCD}} S'$ because $\operatorname{up}^{\Gamma} K \subseteq S'$ and thus $\bigwedge^{\mathsf{FCD}} \operatorname{up}^{\Gamma} K \sqsupseteq \bigwedge^{\mathsf{FCD}} S'$ .