Tiling relation on the set of partitions Let $x\neq \emptyset$ be a set and let $\text{Part}(x)$ be the collection of
all partitions of $x$. We need the following notation. Let $P \in \text{Part}(x)$
and $t\subseteq x$. We set
$$P_{[t]} = \{p\in P:p\cap t \neq \emptyset\}.$$
We define the tiling relation
on $\text{Part}(x)$ by
$$ P \triangleleft Q \textrm{ if and only if for all } S\subseteq 
P\textrm{ we have } \mathsf{card}(S) \leq \mathsf{card}(Q_{[\bigcup S]}).$$
In other words, the relation $P\triangleleft Q$ holds if no subset $S$ of $P$
is covered by a subset of $Q$ having a smaller cardinality than $S$.
Questions. 


*

*Is the relation $\triangleleft$ on $\text{Part}(x)$ transitive?

*For $P,Q \in \text{Part}(x)$ is there $Z\in \text{Part}(x)$ such that

*

*$Z$ refines $P\cup Q$ and

*$Z \triangleleft P$ and $Z\triangleleft Q$?


 A: Question 1. The answer is No. 
Let $x= \{1,2,3,4\}$. For $i\in \{1,2,3\}$ let
$P_i$ be the partition that has $\{i, i+1\}$ as the only non-singleton
partition block. Then $P_i\triangleleft P_{i+1}$ for $i = 1,2$, but it is easy to verify that  $\neg(P_1 \triangleleft P_3)$.
Question 2. I can only answer this in the case that the base set $x$ is finite.  Let $x$ be a set of cardinality $n\in \omega$, and let 
$P, Q \in \text{Part}(x)$.  Consider the set
$$S = \{Y \in \text{Part}(x) : Y \textrm{ refines }P \cup Q\}$$
and set
$$m = \min \{\text{card}(Y) : Y \in S \}.$$
We have $m \leq n$ , and there is $Y_0 \in S$ with $\text{card}(Y_0) = m$. 
Then I
claim that $Y_0 \triangleleft P$ and $Y_0 \triangleleft Q$. 
The reason is if $\neg(Y_0 \triangleleft P)$, say, then you
can find a refinement of $Y_0 \cup P$ (which also refines $P \cup Q$)
that has smaller cardinality than $Y_0$, contradicting the minimality of $\text{card}(Y_0)$.
However I would be interested to see what the answer is to Question 2 for base set $x$ infinite.
