Upgrading a completary multifuncoid

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a set $X$, let $\mathcal F(X)$ be all filters on $\mathcal P(X)$, including the improper filter $\bot_X=\mathcal P(X)$. A filter is a nonempty family closed under finite intersections and supersets. Order filters by $\mathcal A\sqsubseteq\mathcal B$ when $\mathcal A\supseteq\mathcal B$ as families. Thus $\bigsqcup$ is intersection of filter families and $\bigsqcap$ is the filter generated by their union; the empty join is $\bot_X$ and the empty meet is $\top_X=\{X\}$. For $E\subseteq X$, $\uparrow E=\{U\subseteq X:E\subseteq U\}$ is its principal filter. Write $\mathcal A\mathrel{\#}\mathcal B$ when every $U\in\mathcal A$ and $V\in\mathcal B$ have $U\cap V\ne\varnothing$, equivalently $\mathcal A\sqcap\mathcal B\ne\bot_X$.

For an index set $I$ and join-semilattices $L_i$ with bottom $0_i$, their product is ordered coordinatewise. Call $F\subseteq\prod_{i\in I}L_i$ completary when, for every $u,v$ in that product, \[ (u_i\vee v_i)_{i\in I}\in F \quad\Longleftrightarrow\quad \exists c:I\to\{0,1\}:\ (w_i)_{i\in I}\in F, \quad w_i=\begin{cases}u_i&c(i)=0,\\v_i&c(i)=1,\end{cases} \] and no tuple having a bottom coordinate belongs to $F$. This is the source's definition of a completary multifuncoid.Fix any set $X$ and any index set $I$. Let $\mathcal P_f(X)=\{\uparrow A:A\subseteq X\}\subseteq\mathcal F(X)$ be the principal filters, with inherited order. For $u\in\mathcal F(X)^I$, define \[ \operatorname{up}(u)=\{v\in\mathcal P_f(X)^I:u_i\sqsubseteq v_i\text{ for all }i\in I\}, \quad E^*F=\{u\in\mathcal F(X)^I:\operatorname{up}(u)\subseteq F\}. \] Here $\operatorname{up}$ and $E^*$ refer to the fixed filtrator $(\mathcal F(X)^I,\mathcal P_f(X)^I)$.

Statement recorded in the supplied source.

For every $X,I$ and every completary multifuncoid $F\subseteq\mathcal P_f(X)^I$, is its upgraded set $E^*F\subseteq\mathcal F(X)^I$ completary for the filter-lattice joins? The quantifier includes infinite index sets. [S2]

Short English statement

Does upgrading a completary multifuncoid from principal filters to all filters preserve its defining simultaneous join rule, even with infinitely many coordinates?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3427 (OPG-37385). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Upgrading a completary multifuncoid, OPG-37385. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/upgrading_a_completary_multifuncoid.