A note on cut-worthiness of recognizable tree series

The topic of this short note is tree series over semirings with partially ordered carrier set. Thus, tree series become poset-valued fuzzy sets. Starting with a collection of recognizable tree languages we construct a tree series recognizable over any semiring, such that the carrier set is a poset generated by the collection. In the framework of fuzzy structures, the starting collection becomes a subset of the collection of the corresponding cut sets. Under some stricter conditions, it is even equal to this collection. We also partially solve an open problem which was posed in Borchardt et al. [Cut sets as recognizable tree languages, Fuzzy Sets and Systems 157 (2006) 1560–1571]. Namely, we show that if ϕ is a given tree series over a partially ordered, locally finite semiring A, then ϕ is recognizable if and only if there are finitely many cut sets of ϕ and every cut set is a recognizable tree language.