  A classical problem in grammatical inference is to identify a language from a
set of examples. In this paper, we address the problem of identifying a union
of languages from examples that belong to several different unknown languages.
Indeed, decomposing a language into smaller pieces that are easier to represent
should make learning easier than aiming for a too generalized language. In
particular, we consider k-testable languages in the strict sense (k-TSS). These
are defined by a set of allowed prefixes, infixes (sub-strings) and suffixes
that words in the language may contain. We establish a Galois connection
between the lattice of all languages over alphabet {\Sigma}, and the lattice of
k-TSS languages over {\Sigma}. We also define a simple metric on k-TSS
languages. The Galois connection and the metric allow us to derive an efficient
algorithm to learn the union of k-TSS languages. We evaluate our algorithm on
an industrial dataset and thus demonstrate the relevance of our approach.
