Abstract Grammatical inference and finite semigroup theory are continuously developing. Nevertheless it seems that they are not interacting too much. We propose in this paper an inference method for languages that belong to varieties of the form V * LI . Many well known families of languages like locally testable, reversible, dot-depth one, etc. are of that form. The method unifies existing algorithms for inference of some of those families and can be applied to some others that had not been inferred yet. It uses a result about the cascade product in which one of the factors is a transducer and the second is the automaton obtained by inferring the base case of the family using the transduction of the sample as input.