On the definition of clone of relations I am reading A short introduction to clones and I am stuck at this definition ($A$ is a set and $R_A$ the set of finitary relations on $A$)
Definition A subset $R\subseteq R_A$ is called a clone of relations on $A$ if:
(i) $\varnothing\in R$
(ii) $R$ is closed under general superposition, that is, the following holds: for an arbitrary index set $I$, let $\sigma_i\in R^{(k_i)}$ ($i\in I$) and let $\phi:k\longmapsto\alpha$ and $\phi_i:k_i\longmapsto\alpha$ be mappings where $\alpha$ is some cardinal number. Then the relation defined by
$${\displaystyle\bigwedge}^{\phi}_{(\phi_i)_{i\in I}} (\sigma_i)_{i\in I}:=\{r\circ \phi| \forall i\in I: r\circ \phi_i\in \sigma_i,r\in A^{\alpha}\}$$
belongs to $R$.
Here are some questions:
1) What is $k$?
2) What does it mean that $\phi_i$ is a map? What is a map from a number to a number?
3) I know what the composition of two or many relations is, but what is meant here by $r\circ \phi_i$?

Here are some questions:
1) What is $k$?
$k$ is the natural number that is the arity of the relation mentioned in part (ii) of the definition.
2) What does it mean that $\phi_i$ is a map? What is a map from a number to a number?
map=function.
3) I know what the composition of two or many relations is, but what is meant here by $r\circ \phi_i$?
$r\circ  \phi_i$ is the composition of the function $r\colon \alpha\to A$ with the function $\phi_i\colon k_i\to \alpha$. It is therefore a function $k_i\to A$, hence is a potential element of $\sigma_i$, since $\sigma_i\subseteq A^{k_i}$.
