Decomposition of completions of reloids

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 reloid $f:X\to Y$ is a filter on $X\times Y$, ordered by reverse inclusion of filter families. For a relation $R\subseteq X\times Y$, its principal reloid is $\uparrow R$. The reloid $g\circ f:X\to Z$ is generated by $G\circ F$ for $F\in f,G\in g$, where $G\circ F=\{(x,z):\exists y,(x,y)\in F,(y,z)\in G\}$. The reverse $f^{-1}$ is generated by $F^{-1}=\{(y,x):(x,y)\in F\}$ for $F\in f$. The identity $1_X$ is $\uparrow\Delta_X$, where $\Delta_X=\{(x,x):x\in X\}$.For a reloid $f:X\to Y$, define its source-singleton restriction $f|_x$ as the meet of $f$ and $\uparrow(\{x\}\times Y)$. Set \[ \operatorname{Compl}(f)=\bigsqcup_{x\in X}f|_x, \qquad \operatorname{CoCompl}(f)=\left(\operatorname{Compl}(f^{-1})\right)^{-1}. \] Call $f$ complete if $\operatorname{Compl}(f)=f$, and co-complete if $\operatorname{CoCompl}(f)=f$.

Statement recorded in the supplied source.

Do all five identities hold for arbitrary reloids whenever their displayed compositions are defined?

1. $\operatorname{Compl}(g\circ f)=\operatorname{Compl}(g)\circ f$ if $f$ is co-complete.
2. $\operatorname{CoCompl}(f\circ g)=f\circ\operatorname{CoCompl}(g)$ if $f$ is complete.
3. $\operatorname{CoCompl}(\operatorname{Compl}(g)\circ f)=\operatorname{Compl}(g\circ\operatorname{CoCompl}(f))=\operatorname{Compl}(g)\circ\operatorname{CoCompl}(f)$.
4. $\operatorname{Compl}(g\circ\operatorname{Compl}(f))=\operatorname{Compl}(g\circ f)$.
5. $\operatorname{CoCompl}(\operatorname{CoCompl}(g)\circ f)=\operatorname{CoCompl}(g\circ f)$.

In item 2, the typing is $g:Z\to X$ and $f:X\to Y$; in items 1,3,4,5 it is $f:X\to Y$ and $g:Y\to Z$. No common endomorphism assumption is imposed. [S2][S3]

Short English statement

Determine whether completion and co-completion obey all five stated composition identities, retaining the one-sided completeness hypotheses and correct order of composition.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3434 (OPG-56573). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Decomposition of completions of reloids, OPG-56573. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/decomposition_of_completions_of_reloids.
• S3 Victor Porton, General Topology as Ordered Semigroup Actions, author manuscript, version of 21 July 2026. Completion and co-completion; decomposition conjecture containing the five formulas. https://github.com/vporton/agt-combined/blob/759e45cced7769cf3a84ce60ed6c30f4ce9ffa8e/chap-reloids.tex.