Cut sets as recognizable tree languages

A tree series over a semiring with partially ordered carrier set can be considered as a fuzzy set. We investigate conditions under which it can also be understood as a fuzzified recognizable tree language. In this sense, sufficient conditions are presented which, when imposed, ensure that every cut set, i.e., the pre-image of a prime filter of the carrier set, is a recognizable tree language. Moreover, such conditions are also presented for cut sets of recognizable tree series.