multifuncoid

Several ways to apply a (multivalued) multiargument function to a family of filters ★★★

Author(s): Porton

Problem Let $\mathcal{X}$ be an indexed family of filters on sets. Which of the below items are always pairwise equal?

1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters $\mathcal{X}$ .

2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters $\mathcal{X}$ .

3. $\bigcap_{F\in\operatorname{up}^{\mathrm{FCD}}\prod^{\mathrm{Strd}}\mathcal{X}}\langle f \rangle F$ .

Keywords: funcoid; function; multifuncoid; staroid

Posted by porton
updated January 15th, 2020

2 comments

Topology

Graph product of multifuncoids ★★

Author(s): Porton

Conjecture Let $F$ is a family of multifuncoids such that each $F_i$ is of the form $\lambda j \in N \left( i \right) : \mathfrak{F} \left( U_j \right)$ where $N \left( i \right)$ is an index set for every $i$ and $U_j$ is a set for every $j$ . Let every $F_i = E^{\ast} f_i$ for some multifuncoid $f_i$ of the form $\lambda j \in N \left( i \right) : \mathfrak{P} \left( U_j \right)$ regarding the filtrator $\left( \prod_{j \in N \left( i \right)} \mathfrak{F} \left( U_j \right) ; \prod_{j \in N \left( i \right)} \mathfrak{P} \left( U_j \right) \right)$ . Let $H$ is a graph-composition of $F$ (regarding some partition $G$ and external set $Z$ ). Then there exist a multifuncoid $h$ of the form $\lambda j \in Z : \mathfrak{P} \left( U_j \right)$ such that $H = E^{\ast} h$ regarding the filtrator $\left( \prod_{j \in Z} \mathfrak{F} \left( U_j \right) ; \prod_{j \in Z} \mathfrak{P} \left( U_j \right) \right)$ .