In the book Hyperidentities and Clones they (Denecke and Wismath) write:
$xy \approx yx$, in other words
$$F(x,y)=F(y,x)$$
considered as a hyperidentity implies
$$x\approx y.$$
I would like to see a detailed explanation of this observation yielding trivial variety.
I have a fundamental difficulty with hyperidentities due to their recursive nature.
EDIT:
Let $\sigma:\{f_i:i\in I\}\to W_\tau(X)$ be a mapping assigning to every $n_i$-ary operation symbol $f_i$ of type $\tau$ an $n_i$-ary term, $\sigma(f_i)$. Any such mapping $\sigma$ will be called a hypersubstitution of type $\tau$.
Here $W_\tau(X)$ is the usual recursive definition of terms:
$x_1,...,x_n$ are $n$-ary terms
if $w_1,...,w_m$ are $n$-ary terms and $m=n_i$ (for some $i\in I$) then $f_i(w_1,...,w_m)$ is an $n$-ary term.
NOW we can think of any hypersubstitution $\sigma$ as mapping the term $f_i(x_1,...,x_{n_i})$ to the term $\sigma(f_i)$. It follows that every hypersubstitution of type $\tau$ induces a mapping $\hat{\sigma}:W_\tau(X)\to W_\tau(X)$ as follows:for any $w\in W_\tau(X)$, the term $\hat{\sigma}[w]$ is defined by
(1) $\hat{\sigma}[x]:=x$ for any variable $x\in X$
(2) $\hat{\sigma}[f_i(w_1,...,w_{n_i})]:=\sigma(f_i)(\hat{\sigma}[w_1],...,\hat{\sigma}[w_{n_i}]).$
So now they write: we see that an hypersubstitution of the binary term $x$ for the operation symbol $F$ (in other words,application of the hypersubstitution $\sigma_x$) yields the identity $x\approx y$.
Here I do not even understand how a single variable $x$ should be a binary term.Nor do I understand how the recursive definition should be applied to $F$.