Infinite distributivity of meet over join for a principal funcoid

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 $\mathcal A\sqcap\mathcal B\ne\bot_X$.

A funcoid $f:X\to Y$ is a pair of maps $\alpha_f:\mathcal F(X)\to\mathcal F(Y)$ and $\beta_f:\mathcal F(Y)\to\mathcal F(X)$ satisfying \[ \mathcal B\mathrel{\#}\alpha_f(\mathcal A) \quad\Longleftrightarrow\quad \mathcal A\mathrel{\#}\beta_f(\mathcal B) \qquad(\mathcal A\in\mathcal F(X),\ \mathcal B\in\mathcal F(Y)). \] Its reverse interchanges $\alpha_f,\beta_f$. Composition satisfies $\alpha_{g\circ f}=\alpha_g\circ\alpha_f$ and $\beta_{g\circ f}=\beta_f\circ\beta_g$. The identity has both maps equal to the identity. The order on $\mathsf{FCD}(X,Y)$ is pointwise order on $\alpha_f$; its meets and joins mean greatest lower bounds and least upper bounds within this complete lattice. A relation $R\subseteq X\times Y$ induces the principal funcoid $\widehat R$ with $\alpha_{\widehat R}(\mathcal A)$ the filter generated by $\{R[U]:U\in\mathcal A\}$ and the analogous inverse-image formula for $\beta$. Here $R[U]=\{y:\exists x\in U,(x,y)\in R\}$.

Statement recorded in the supplied source.

For arbitrary sets $X,Y$, every principal funcoid $f=\widehat R\in\mathsf{FCD}(X,Y)$ and every set $\mathcal S\subseteq\mathsf{FCD}(X,Y)$, including $\mathcal S=\varnothing$, does \[ f\sqcap\bigsqcup\mathcal S =\bigsqcup_{g\in\mathcal S}(f\sqcap g) \] hold? The notation $\langle f\sqcap\rangle^*\mathcal S$ in the source is the image family $\{f\sqcap g:g\in\mathcal S\}$. [S2][S3]

Short English statement

Does meet with a principal funcoid distribute over every join in the same funcoid lattice?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3441 (OPG-60017). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Infinite distributivity of meet over join for a principal funcoid, OPG-60017. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/infinite_distributivity_of_meet_over_join_for_a_principal_funcoid.
• S3 Victor Porton, General Topology as Ordered Semigroup Actions, author manuscript, version of 21 July 2026. Basic definitions and infinite-distributivity conjecture immediately after finite-distributivity proof. https://github.com/vporton/agt-combined/blob/759e45cced7769cf3a84ce60ed6c30f4ce9ffa8e/chap-funcoids.tex.