# distributivity

## Infinite distributivity of meet over join for a principal funcoid ★★

Author(s): Porton

\begin{conjecture} $f \sqcap \bigsqcup S = \bigsqcup \langle f \sqcap \rangle^{\ast} S$ for principal funcoid $f$ and a set $S$ of funcoids of appropriate sources and destinations. \end{conjecture}

Keywords: distributivity; principal funcoid

## Distributivity of a lattice of funcoids is not provable without axiom of choice ★

Author(s): Porton

\begin{conjecture} Distributivity of the lattice $\mathsf{FCD}(A;B)$ of funcoids (for arbitrary sets $A$ and $B$) is not provable in ZF (without axiom of choice). \end{conjecture}

A similar conjecture:

\begin{conjecture} $a\setminus^{\ast} b = a\#b$ for arbitrary filters $a$ and $b$ on a powerset cannot be proved in ZF (without axiom of choice). \end{conjecture}

## Distributivity of inward reloid over composition of funcoids ★★

Author(s): Porton

\begin{conjecture} $( \mathsf{\tmop{RLD}})_{\tmop{in}} (g \circ f) = ( \mathsf{\tmop{RLD}})_{\tmop{in}} g \circ ( \mathsf{\tmop{RLD}})_{\tmop{in}} f$ for any composable \href[funcoids]{http://www.wikinfo.org/index.php/Funcoid} $f$ and $g$. \end{conjecture}

## Distributivity of outward reloid over composition of funcoids ★★

Author(s): Porton

\begin{conjecture} $( \mathsf{\tmop{RLD}})_{\tmop{out}} (g \circ f) = ( \mathsf{\tmop{RLD}})_{\tmop{out}} g \circ ( \mathsf{\tmop{RLD}})_{\tmop{out}} f$ for any composable \href[funcoids]{http://www.wikinfo.org/index.php/Funcoid} $f$ and $g$. \end{conjecture}