Hyperidentities and solid varieties (Q861607)

Consider a fixed type of algebras. An identity \(u=v\) of a variety \(V\) is called a hyperidentity of \(V\) if every identity obtained from \(u=v\) by replacing every fundamental operation symbol occurring in \(u=v\) by a term of the same arity (the same fundamental operation symbols are replaced by the same terms) also holds in \(V\). \(V\) is called solid if every identity holding in \(V\) is a hyperidentity. The solid varieties of a fixed type form a complete sublattice of the lattice of all varieties of this type. An overview (with results and ideas of the corresponding proofs) on hyperidentities and solid varieties in the case of semigroup varieties, varieties of type \((n)\) and varieties of arbitrary type is given, culminating in the construction of an infinite chain of solid varieties for an arbitrarily given type.