User:DaNASCAT/Sandbox

Prerequisites: Algebraic General Topology.

Conjecture. $L\in {\mathrel {\left[f\right]}}\Rightarrow {\mathrel {\left[f\right]}}\cap \prod _{i\in \operatorname {dom} {\mathfrak {A}}}\operatorname {atoms} L_{i}\neq \emptyset$ for every pre-multifuncoid $f$ of the form whose elements are atomic posets.

A weaker conjecture: It is true for forms whose elements are powersets.

The following is an attempted proof:

If $\operatorname {arity} f=0$ our theorem is trivial, so let $\operatorname {arity} f\neq 0$. Let $\sqsubseteq$ is a well-ordering of $\operatorname {arity} f$ with greatest element $m$.

Let $\Phi$ is a function which maps non-least elements of posets into atoms under these elements and least elements into themselves. (Note that $\Phi$ is defined on least elements only for completeness, $\Phi$ is never taken on a least element in the proof below.) {\color{brown} [TODO: Fix the universal set paradox here.]}

Define a transfinite sequence $a$ by transfinite induction with the formula Failed to parse (syntax error): {\displaystyle a_c = \Phi \left\langle f \right\rangle_c  \left( a|_{X \left( c \right) \setminus \left\{ c \right\}} \cup L|_{\left( \operatorname{arity} f \right) \setminus X \left( c \right)} \right)} .

Let $b_{c}=a|_{X\left(c\right)\setminus \left\{c\right\}}\cup L|_{\left(\operatorname {arity} f\right)\setminus X\left(c\right)}$. Then $a_{c}=\Phi \left\langle f\right\rangle _{c}b_{c}$.

Let us prove by transfinite induction $a_{c}\in \operatorname {atoms} L_{c}.$ $a_{c}=\Phi \left\langle f\right\rangle _{c}L|_{\left(\operatorname {arity} f\right)\setminus \left\{c\right\}}\sqsubseteq \left\langle f\right\rangle _{c}L|_{\left(\operatorname {arity} f\right)\setminus \left\{c\right\}}$. Thus $a_{c}\sqsubseteq L_{c}$. [TODO: Is it true for pre-multifuncoids?]

The only thing remained to prove is that $\left\langle f\right\rangle _{c}b_{c}\neq 0$

that is Failed to parse (syntax error): {\displaystyle \langle f \rangle _ c  ( a|_{ X ( c ) \setminus \{ c \} } \cup L|_{( \operatorname{arity} f ) \setminus X ( c )} ) \neq 0} that is $y\not \asymp \left\langle f\right\rangle _{c}b_{c}$.