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 $F$.

  • This is the unique question on this site about hyperidentities and I guess very few readers will know the corresponding definition. Please give a link (or a hyperlink...) or include the full definition in your question. – J.-E. Pin Jan 4 at 18:04
  • I've just tried to send you an email to pin@litp.ibp.fr but it couldn't be delivered.Do you please have the correct email address? You can reach me at (pax0@seznam.cz) My question is from the book Hyperidentities and clones from the page 71. Do you have it at hand?It can be found via. google books, but unfortunately the relevant page could not be shown. – user122424 Jan 4 at 18:28
  • It was not a personal demand but just an advice if you wish to have an answer some day. – J.-E. Pin Jan 4 at 22:29
  • And could you please answer this question? – user122424 Jan 5 at 12:59

Let $t(x,y)=x$ ($x$ as a binary term). If $F(x,y) \approx F(y,x)$ is hyper-satisfied by an algebra, then, in that algebra we get (hyper-substituting $t$ for $F$) $$t(x,y) \approx t(y,x),$$ whence $$x \approx y.$$

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