On groups of hypersubstitutions (Q616120)

Let \(\tau : \Omega \to \mathbb{N}\) be a finitary type, where \(\Omega\) is a set of symbols of algebraic operations and \(\tau (\omega )\) is the arity of the \(\omega\). Let \(P\) denote the ordered set of positive integers and \(P\Omega \) denote the set of \(\Omega\)-words over \(P\). The so-called derived type \(\tau': P\Omega \to \mathbb{N}\) was defined in [\textit{J. D. H. Smith} and \textit{A. B. Romanowska}, Post-modern algebra. New York, NY: Wiley (1999; Zbl 0946.00001), \S IV 1.3]. A hypersubstitution of type \(\tau \) is a morphism \(\sigma : \tau \to \tau '\) in the slice category \(\text{Set}/N\), that is, a function \(\sigma : \Omega \to P\Omega \) such that \(\tau ' \circ \sigma =\tau\). Each hypersubstitution \(\sigma : \Omega \to P\Omega \) extends to a corresponding function \(\hat{\sigma} : P\Omega \to P\Omega \). Let \(\mathcal V\) be a variety of algebras of type \(\tau\). A hypersubstitutions \(\sigma\) is called (\(\mathcal V\)-)proper if \(\hat{\sigma} (u)=\hat{\sigma} (v) \) for every identity \(u=v\) of \(\mathcal V\). In the paper under review, the proper hypersubstitutions of a variety \(\mathcal V\) of the kind \(\sigma : \Omega \to \Omega \) are considered. If hypersubstitution \(\sigma : \Omega \to \Omega \) is invertible and (\(\mathcal V \)-)proper, it is called a \(\mathcal V\)-hyperequivalence. In particular, it is proved that every monoid may be realized as a monoid of hypersubstitutions and every automorphism group of a monoid is isomorphic to the \(\mathcal V\)-hyperequivalence group of some variety \(\mathcal V\). Since groups of hyperequivalences can be regarded as symmetries of algebraic theories, the author takes an interest in the phenomenon of symmetry-breaking. He considers lattice theory and quasigroup theory from this point of view.