Can a $k$-ary relation have polymorphisms of arity greater than $k$?
Let's say that there's a [$k$-ary] relation $R$ and [$m$-ary] function $f$ such that $m&gt;k$. Is $f$ a polymorphism of $R$?


Maybe, maybe not: it depends on the function and the relation.  A given $k$-ary relation may have polymorphisms of arity arity less than, equal to, and/or greater than&nbsp;$k$. In fact, every relation has polymorphisms of all arities. For example, the ($1$-ary) identity function is a polymorphism of every relation, as are the projection functions $f_\ell(x_1, \dots, x_m) = x_\ell$ for all $1\leq \ell\leq m$. (Exercise: check these examples.) Notice that the identity function is just the case $\ell=m=1$.


  Concrete example: Assuming an $\wedge$ relation $\{ (1,1) \}$ (i.e. arity is 2), is the majority function $f(a,b,c) = (a \wedge b) \vee (a \wedge c) \vee (b \wedge c)$ a polymorphism of this relation?


Check the definition! $f$&nbsp;is a polymorphism of&nbsp;$R_\land = \{(1,1)\}$ if, and only if, for all $(x_1,x_2),(y_1,y_2),(z_1,z_2)\in R_\land$, we have $(f(x_1,y_1,z_1),f(x_2,y_2,z_2))\in R_\land$. Well, the only tuple in&nbsp;$R_\land$ is $(1,1)$, so we only care about the case $x_1=y_1=z_1=x_2=y_2=z_2=1$. And we see that $f(1,1,1)=1$, so $(f(1,1,1),f(1,1,1))\in R_\land$, so $f$&nbsp;is, indeed, a polymorphism of $R_\land$. Further, by exactly the same argument, any function such that $f(1, \dots, 1) = 1$ is a polymorphism of&nbsp;$R_\land$.

The polymorphism property is often represented as a table: given a $k$-ary relation&nbsp;$R$ and an $m$-ary function&nbsp;$f$, we can write

$$\begin{matrix}
    (t_{1,1} &amp; t_{1,2} &amp; \cdots &amp; t_{1,k}) &amp; \in R\\
    (t_{2,1} &amp; t_{2,2} &amp; \cdots &amp; t_{2,k}) &amp; \in R\\
    \vdots   &amp; \vdots  &amp; \ddots &amp; \vdots   &amp; \\
    (t_{m,1} &amp; t_{m,2} &amp; \cdots &amp; t_{m,k}) &amp; \in R\\
\hline
    (f(t_{1,1}, \dots, t_{m,1}) &amp; f(t_{1,2}, \dots, t_{m,2}) &amp;\cdots &amp; f(t_{1,k}, \dots, t_{m,k})) &amp;\in R
\end{matrix}$$

The meaning is that we require that, whenever the tuples above the line are in&nbsp;$R$, the tuple below the line must also be in&nbsp;$R$.  Notice that nothing in the structure of the table requires any relation between $k$ and&nbsp;$m$: the property could hold (or not hold) for a relation of any arity and a function of any arity.  It's just required that, whenever we write a collection of tuples in&nbsp;$R$ and "apply&nbsp;$f$ to each column", the resulting tuple is always still in&nbsp;$R$.