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

\begin{conjecture} The poset of multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$: \begin{enumerate} \item atomic; \item atomistic. \end{enumerate} \end{conjecture}

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

Refer to \href[this Web site]{http://www.mathematics21.org/algebraic-general-topology.html} for the theory which I now attempt to generalize.

\begin{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 \[ \forall A, B \in \mathfrak{A}: \left( A \cup B \in S \Leftrightarrow A \in S \vee B \in S \right) . \] \end{definition}

\begin{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 \[ \left( \ensuremath{\operatorname{val}}f \right)_i L = \left\{ X \in \mathfrak{A}_i \hspace{0.5em} | \hspace{0.5em} L \cup \left\{ (i ; X) \right\} \in f \right\} . \] \end{definition}

\begin{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: \begin{enumerate} \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. \end{enumerate} \end{definition}

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

Bibliography

* \href [Algebraic General Topology]{http://www.mathematics21.org/algebraic-general-topology.html}

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)


* indicates original appearance(s) of problem.