
Atomicity of the poset of multifuncoids
Conjecture The poset of multifuncoids of the form
is for every sets
and
:



- \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
with least element 0 is a set
such that
and



![\[ \forall A, B \in \mathfrak{A}: \left( A \cup B \in S \Leftrightarrow A \in S \vee B \in S \right) . \]](/files/tex/70420d16ddf609e4c505908182520a5bcf379d3e.png)
Definition Let
be a family of posets,
(
has the order of function space of posets),
,
. Then





![\[ \left( \ensuremath{\operatorname{val}}f \right)_i L = \left\{ X \in \mathfrak{A}_i \hspace{0.5em} | \hspace{0.5em} L \cup \left\{ (i ; X) \right\} \in f \right\} . \]](/files/tex/402ce92b70bfd908eefa69f8ec7f3b5cd3cb72d2.png)
Definition Let
is a family of posets. A multidimensional funcoid (or multifuncoid for short) of the form
is an
such that we have that:



- \item



\item is an upper set.
is a function space over a poset
that is
for
.
Bibliography
* indicates original appearance(s) of problem.