Let $A$ be a set. A partition of $A$ is a collection $\pi$ of nonempty subsets of $A$ such that
- $\bigcup\pi = A$, and
- $x\cap y = \varnothing$ for distinct $x$ and $y$ in $\pi$.
Let $P(A)$ be the collection of partitions of $A$, and let $\lesssim$ be the refinement order on $P(A)$, which for any $\pi$ and $\sigma$ in $P(A)$ is defined by $\pi\lesssim\sigma$ if and only if for each $x$ in $\pi$ there is a $y$ in $\sigma$ such that $x\subseteq y$.
$P(A)$ ordered by refinement is a lattice, and an explicit construction of the join $\pi\vee\sigma$ of any two partitions $\pi$ and $\sigma$ in $P(A)$ has been discussed here: call any blocks $x$ and $y$ in $\pi$ or $\sigma$ equivalent if and only if there are $w_1,\ldots,w_n$ in $\pi\cup\sigma$ such that $x = w_1$ and $y = w_n$ and $w_i\cap w_{i+1}\neq\varnothing$ for $i=1,\ldots,i-1$, then this defines an equivalence relation $\equiv$ on $\pi\cup\sigma$ and each block in $\pi\vee\sigma$ equals the union of an equivalence class of $\equiv$.
This explicit construction of the join of two partitions extends to one for arbitrary nonempty subsets $S$ of $P(A)$ by allowing the $w_1,\ldots,w_n$ to be elements in $\bigcup S$.
Question: Does this explicit construction of the join of a subset $S$ of $P(A)$ extend to the case where $S$ is empty, and if so, how?
It is well-known that $P(A)$ ordered by refinement is a complete lattice, so the join $\bigvee\varnothing$ of the empty set exists, and that it equals the partition consisting of singleton sets $\{a\}$, $a\in A$.
For nonempty subsets $S$ of $P(A)$, applying the above construction, I can write a block in $\bigvee S$ as the union of the equivalence class $[\pi]_\equiv$ of an element $\pi$ in $\bigcup S$. But if $S$ is empty, then there are no $\pi$ in $\bigcup S$ to construct an equivalence class $[\pi]_\equiv$ for. Is there a way to make the above construction work when $S$ is empty?
It seems to me, now, that this can be done as follows.
For each $a$ in $A$, let $[a]$ be the collection of sets $\{a\}$ and all $w$ in $\bigcup S$ for which there are $w_1,\ldots,w_n$ in $\bigcup S$, where $n$ is a positive integer, such that $w_1 = \{a\}$, $w_i\cap w_{i+1}$ is nonempty, and $w_n = w$. Then $\bigvee S$ is the collection of sets $\bigcup[a]$, $a\in A$.