# Graph product of multifuncoids

 Importance: Medium ✭✭
 Author(s): Porton, Victor
 Subject: Topology
 Keywords: graph-product multifuncoid
\begin{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 $\left( \prod_{j \in Z} \mathfrak{F} \left( U_j \right) ; \prod_{j \in Z} \mathfrak{P} \left( U_j \right) \right)$. \end{conjecture}
See \href [Algebraic General Topology]{http://www.mathematics21.org/algebraic-general-topology.html}, especially \href [the theory of multifuncoids]{http://www.mathematics21.org/binaries/nary.pdf} for definitions of used concepts. % You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}