User:DaNASCAT/Sandbox

Prerequisites: Algebraic General Topology.

Conjecture. $L\in \left[f\right]\Rightarrow \left[f\right]\cap {\prod }_{i\in \mathrm{d}\mathrm{o}\mathrm{m}\mathfrak{A}}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}{L}_i\ne \mathrm{\varnothing }$ 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 $\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}f=0$ our theorem is trivial, so let $\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}f\ne 0$. Let $⊑$ is a well-ordering of $\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}f$ with greatest element $m$.

Let $\mathrm{\Phi }$ is a function which maps non-least elements of posets into atoms under these elements and least elements into themselves. (Note that $\mathrm{\Phi }$ is defined on least elements only for completeness, $\mathrm{\Phi }$ 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 $b_c=a{|}_{X\left(c\right)\setminus \left\{c\right\}}\cup L{|}_{\left(\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}f\right)\setminus X\left(c\right)}$. Then $a_c=\mathrm{\Phi }{⟨f⟩}_c{b}_c$.

Let us prove by transfinite induction $a_c\in \mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}{L}_c.$ $a_c=\mathrm{\Phi }{⟨f⟩}_{c}L{|}_{\left(\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}f\right)\setminus \left\{c\right\}}⊑{⟨f⟩}_{c}L{|}_{\left(\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}f\right)\setminus \left\{c\right\}}$. Thus $a_c⊑L_c$. [TODO: Is it true for pre-multifuncoids?]

The only thing remained to prove is that ${⟨f⟩}_c{b}_c\ne 0$

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 $y\asymp ̸{⟨f⟩}_c{b}_c$.