Abstract
We show: There is no tree language whose syntactic semigroup lies in Kr, or in LI unless it is in K. Instead, reverse (resp. general) definite tree languages have syntactic semigroups in Kr V J1 (resp. LI v J1). We give concise combinatorial descriptions of the tree languages whose syntactic semigroups are in Krv J1 (resp. LI v J1), in terms of the properties reverse (resp. general) J1-definite. These properties are more general than the properties reverse (general) definite as defined by Heuter. Finally we show that they are decidable.
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