Fuzzy tree language recognizability

A fuzzy tree language with membership grades in an arbitrary set is syntactically recognizable (s-recognizable) if its syntactic algebra is finite. The equality problem for such languages is decidable and their syntactic algebra can be effectively constructed provided that they are s-recognizable. Linear (but non arbitrary) tree homomorphisms preserve s-recognizability. Tree automata whose transitions are weighted over the unit interval and whose behavior is computed with respect to a pair made of a t-norm distributive over a t-conorm have the syntactic recognition power and thus their equivalence problem is decidable. However, s-recognizability is more powerful when dealing with non-distributive pairs of such operations.