  Techniques are developed for creating new and general language families of
only semilinear languages, and for showing families only contain semilinear
languages. It is shown that for language families L that are semilinear full
trios, the smallest full AFL containing L that is also closed under
intersection with languages in NCM (where NCM is the family of languages
accepted by NFAs augmented with reversal-bounded counters), is also semilinear.
If these closure properties are effective, this also immediately implies
decidability of membership, emptiness, and infiniteness for these general
families. From the general techniques, new grammar systems are given that are
extensions of well-known families of semilinear full trios, whereby it is
implied that these extensions must only describe semilinear languages. This
also implies positive decidability properties for the new systems. Some
characterizations of the new families are also given.
