Atomicity of the poset of multifuncoids

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Posted	by:	porton
	on:	February 12th, 2012

Conjecture The poset of multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$ :

\item atomic; \item atomistic.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Definition A free star on a join-semilattice $\mathfrak{A}$ with least element 0 is a set $S$ such that $0 \not\in S$ and $\forall A, B \in \mathfrak{A}: \left( A \cup B \in S \Leftrightarrow A \in S \vee B \in S \right) .$

Definition Let $\mathfrak{A}$ be a family of posets, $f \in \mathscr{P} \prod \mathfrak{A}$ ( $\prod \mathfrak{A}$ has the order of function space of posets), $i \in \ensuremath{\operatorname{dom}}\mathfrak{A}$ , $L \in \prod \mathfrak{A}|_{\left( \ensuremath{\operatorname{dom}}\mathfrak{A} \right) \setminus \left\{ i \right\}}$ . Then $\left( \ensuremath{\operatorname{val}}f \right)_i L = \left\{ X \in \mathfrak{A}_i \hspace{0.5em} | \hspace{0.5em} L \cup \left\{ (i ; X) \right\} \in f \right\} .$

Definition Let $\mathfrak{A}$ is a family of posets. A multidimensional funcoid (or multifuncoid for short) of the form $\mathfrak{A}$ is an $f \in \mathscr{P} \prod \mathfrak{A}$ such that we have that:

$\left( \tmop{val} f \right)_i L$

$i \in \tmop{dom} \mathfrak{A}$

$L \in \prod \mathfrak{A}|_{\left( \tmop{dom} \mathfrak{A} \right) \setminus \left\{ i \right\}}$

\item $f$ is an upper set.

$\mathfrak{A}^n$ is a function space over a poset $\mathfrak{A}$ that is $a\le b\Leftrightarrow \forall i\in n:a_i\le b_i$ for $a,b\in\mathfrak{A}^n$ .