Multifuncoid has atomic arguments

A counter-example against this conjecture have been found. See \hrefhttp://www.mathematics21.org/algebraic-general-topology.htmlAlgebraic General Topology.

Prerequisites: \hrefhttp://www.mathematics21.org/algebraic-general-topology.htmlAlgebraic 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 (partial) 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.) \colorbrown [TODO: Fix the ”universal set” paradox here.]

Define a transfinite sequence $a$ by transfinite induction with the formula $a_{c}=\Phi\left\langle f\right\rangle_{c}\left(a|_{X\left(c\right)\setminus% \left\{c\right\}}\cup L|_{\left(\operatorname{arity}f\right)\setminus X\left(c% \right)}\right)$ .

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 $\langle f\rangle_{c}(a|_{X(c)\setminus\{c\}}\cup L|_{(\operatorname{arity}f)% \setminus X(c)})\neq 0$ that is $y\not\asymp\left\langle f\right\rangle_{c}b_{c}$ .

Title	Multifuncoid has atomic arguments
Canonical name	MultifuncoidHasAtomicArguments
Date of creation	2014-12-14 21:09:54
Last modified on	2014-12-14 21:09:54
Owner	porton (9363)
Last modified by	porton (9363)
Numerical id	2
Author	porton (9363)
Entry type	Conjecture
Classification	msc 54J05
Classification	msc 54A05
Classification	msc 54D99
Classification	msc 54E05
Classification	msc 54E17
Classification	msc 54E99