Atomicity of the poset of completary multifuncoids

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Posted	by:	porton
	on:	February 12th, 2012

Conjecture The poset of completary 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 Let $\mathfrak{A}$ is a family of join-semilattice. A completary multifuncoid of the form $\mathfrak{A}$ is an $f \in \mathscr{P} \prod \mathfrak{A}$ such that we have that:

$L_0 \cup L_1 \in f \Leftrightarrow \exists c \in \left\{ 0, 1 \right\}^n : \left( \lambda i \in n : L_{c \left( i_{} \right)} i \right) \in f$

$L_0, L_1 \in \prod \mathfrak{A}$

\item If $L \in \prod \mathfrak{A}$ and $L_i = 0^{\mathfrak{A}_i}$ for some $i$ then $\neg f L$ .

$\mathfrak{A}^n$ 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 $a,b\in\mathfrak{A}^n$ .