Atomicity of the poset of completary multifuncoids

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Topology
Keywords: multifuncoid
Recomm. for undergrads: no
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:
    \item $ 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 $ for every $ 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 $.

* indicates original appearance(s) of problem.