Hyperassociative varieties of semigroups
Let \(\tau = (n_ i)_{i\in I}\) be a type and \((f_ i)_{i \in I}\) operation symbols such that the arity of \(f_ i\) is \(n_ i\). Let \(W_ \tau(X)\) be the set of all terms of type \(\tau\) in an alphabet \(X\). A map \(\sigma: (f_ i)_{i \in I} \to W_ \tau(X)\) is a hypersubstitution, and \(\sigma\) can be extended to a map \(\widehat{\sigma}: W_{\tau} \to W_{\tau}\) in an obvious way. We then have that \(t \approx t'\) is a hyperidentity of an algebra \(A\) if \(\widehat{\sigma}(t) \approx \widehat{\sigma}(t')\) is an identity in \(A\). An equation is a hyperidentity of a variety \(V\) if it is a hyperidentity of every algebra in \(V\). A variety \(V\) of semigroups is hyperassociative if \(F(F(x,y),z)\approx F(x,F(y,z))\) is a hyperidentity in \(V\). It is easily checked that any hyperassociative variety of semigroups satisfies the identities: \(x^ 4 \approx x^ 2\) and \(xyxzxyx \approx xyzyx\). This paper gives a very much more complicated (but still finite) set of identities which a variety satisfies if and only if it is hyperassociative.
scientific article; zbMATH DE number 611113