Multifuncoid has atomic arguments

A counter-example against this conjecture have been found. See \href General Topology.

Prerequisites: \href General Topology.

Conjecture. L[f][f]idom𝔄atomsLi 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 arityf=0 our theorem is trivial, so let arityf0. Let is a well-ordering of arityf with greatest element m.

Let Φ is a function which maps non-least elements of posets into atoms under these elements and least elements into themselves. (Note that Φ is defined on least elements only for completeness, Φ is never taken on a least element in the proof below.) \colorbrown [TODO: Fix the ”universal setparadoxMathworldPlanetmath here.]

Define a transfinite sequence a by transfinite inductionMathworldPlanetmath with the formulaMathworldPlanetmathPlanetmath ac=Φfc(a|X(c){c}L|(arityf)X(c)).

Let bc=a|X(c){c}L|(arityf)X(c). Then ac=Φfcbc.

Let us prove by transfinite induction acatomsLc. ac=ΦfcL|(arityf){c}fcL|(arityf){c}. Thus acLc. [TODO: Is it true for pre-multifuncoids?]

The only thing remained to prove is that fcbc0

that is fc(a|X(c){c}L|(arityf)X(c))0 that is yfcbc.

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