Summary: Let \(\tau : F \to \mathbb N\) be a type of algebras where \(F\) is the set of fundamental algebras.
A mapping \(\eta \) is called a hypersubstitution if \(\eta \) satisfies the following conditions:
(H1) It assigns to every fundamental term \(f(x_0,\dots ,x_{\tau (f)-1)})\) a term \(\varphi _{f,\tau }(x_0,\dots ,x_{\tau (f)-1})\) and \(\eta (f(x_0,\dots ,x_{\tau (f)-1}))=\varphi _{f,\tau }(x_o,\dots ,x_{\tau (f)-1})\);
(H2) \(\eta (x_k) = x_k\) for every variable \(x_k,0\leq k<\omega \);
(H3) If \(f\in F\) and \(\varphi _0,\dots ,\varphi _{\tau (f)-1}\) then \(\eta (f(\varphi _0,\dots ,\varphi _{\tau (f)-1))} = \varphi _{f,\tau }(\eta (\varphi _0),\dots ,\eta (\varphi _{\tau (f)-1})\).
A hypersubstitution \(\eta \) on type \(\tau \) is called proper if for every identity \(\varphi \approx \psi \) in \(V\) satisfied the identity \(\eta (\varphi) = \eta (\psi)\) is satisfied in \(V\) as well.
An identity \(\varphi \approx \psi \) of type \(\tau \) is callled uniform if \(F(\varphi) = F(\psi) = F\) or \(F(\varphi) = F(\psi) \neq F\) and Var\((\varphi)\)=Var\((\psi)\).
An identity \(\varphi \approx \psi \) of type \(\tau \) is called biregular if \(F(\varphi)=F(\psi)\) and Var\((\varphi)\)= Var\((\psi)\). The author gives various sufficient conditions under which a hypersubstitution \(\eta \) is a proper hypersubstitution of the uniformations and of the biregularization of a variety \(V\).