Atomicity of the poset of 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 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:
    \item $ \left( \tmop{val} f \right)_i L $ is a free star for every $ 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 $.

* indicates original appearance(s) of problem.