Let $A$ be a finite set, $k\in\mathbb N$, $\tilde A\subset A^k$ and $\phi: \tilde A\to A$ be a map.

Consider the following property a set partition $P$ of $A$ might have: $$ \forall B_1,\dotsc,B_k\in P, (b_1,\dotsc,b_k)\in (B_1\times\dotsb\times B_k)\cap\tilde A\; \exists B\in P: \phi(b_1,\dotsc,b_k)\in B. $$ Put differently: the image of $\phi(B_1,\dotsc,B_k)$ is contained in a single block.

I would like to know whether this notion has (in some sense) a name, and whether there is a nice way to compute, given a set partition $Q$ of $A$, the finest set partition coarser than $Q$ having the above mentioned property.

Let $A$ be a finite set, $k\in\mathbb N$, $\tilde A\subset A^k$ and $\phi: \tilde A\to A$ be a map.

Consider the following property a set partition $P$ of $A$ might have: $$ \forall B_1,\dotsc,B_k\in P\; \exists B\in P\; \forall (b_1,\dotsc,b_k)\in (B_1\times\dotsb\times B_k)\cap\tilde A :\phi(b_1,\dotsc,b_k)\in B. $$ Put differently: the image of $\phi(B_1,\dotsc,B_k)$ is contained in a single block.

I would like to know whether this notion has (in some sense) a name, and whether there is a nice way to compute, given a set partition $Q$ of $A$, the finest set partition coarser than $Q$ having the above mentioned property.

Let $A$ be a finite set, $k\in\mathbb N$, $\tilde A\subset A^k$ and $\phi: \tilde A\to A$ be a map.

Consider the following property a set partition $P$ of $A$ might have: $$ \forall B_1,\dots,B_k\in P, (b_1,\dots,b_k)\in (B_1\times\dots\times B_k)\cap\tilde A\; \exists B\in P: \phi(b_1,\dots,b_k)\in B. $$ Put differently: the image of $\phi(B_1,\dots,B_k)$ is contained in a single block.

I would like to know whether this notion has (in some sense) a name, and whether there is a nice way to compute, given a set partition $Q$ of $A$, the finest set partition coarser than $Q$ having the above mentioned property.

Let $A$ be a finite set, $k\in\mathbb N$, $\tilde A\subset A^k$ and $\phi: \tilde A\to A$ be a map.

Consider the following property a set partition $P$ of $A$ might have: $$ \forall B_1,\dotsc,B_k\in P, (b_1,\dotsc,b_k)\in (B_1\times\dotsb\times B_k)\cap\tilde A\; \exists B\in P: \phi(b_1,\dotsc,b_k)\in B. $$ Put differently: the image of $\phi(B_1,\dotsc,B_k)$ is contained in a single block.

I would like to know whether this notion has (in some sense) a name, and whether there is a nice way to compute, given a set partition $Q$ of $A$, the finest set partition coarser than $Q$ having the above mentioned property.

Let $A$ be a finite set, $k\in\mathbb N$, $\tilde A\subset A^k$ and $\phi: \tilde A\to A$ be a map.

Consider the following property a set partition $P$ of $A$ might have: $$ \forall B_1,\dots,B_k\in P, b_i\in B_i\; \exists B\in P: \phi(b_1,\dots,b_k)\in B. $$ Put differently: the image of $\phi(B_1,\dots,B_k)$ is contained in a single block.

I would like to know whether this notion has (in some sense) a name, and whether there is a nice way to compute, given a set partition $Q$ of $A$, the finest set partition coarser than $Q$ having the above mentioned property.

Let $A$ be a finite set, $k\in\mathbb N$, $\tilde A\subset A^k$ and $\phi: \tilde A\to A$ be a map.

Consider the following property a set partition $P$ of $A$ might have: $$ \forall B_1,\dots,B_k\in P, (b_1,\dots,b_k)\in (B_1\times\dots\times B_k)\cap\tilde A\; \exists B\in P: \phi(b_1,\dots,b_k)\in B. $$ Put differently: the image of $\phi(B_1,\dots,B_k)$ is contained in a single block.

I would like to know whether this notion has (in some sense) a name, and whether there is a nice way to compute, given a set partition $Q$ of $A$, the finest set partition coarser than $Q$ having the above mentioned property.