  This paper is devoted to the study of the relative Lipschitz saturation of
complex algebraic varieties. More precisely, we investigate the concept of
Lipschitz saturation of a variety in another, and we focus on the case where
the dominant morphism between the two varieties is not necessarily finite. In
particular, we answer, in the case of algebraic varieties, an open question of
Pham and Teissier concerning the finiteness of the Lipschitz saturation of
general algebras. Finally, we use the Lipschitz saturation to provide algebraic
criteria for two algebraic varieties to be linked by an algebraic morphism,
which is a locally biLipschitz homeomorphism on the closed points of the
variety.
