1 Introduction and preliminaries 

1 






2 Manysorted variety theorem 

9 



2.1 Manysorted algebras 

10 
2.2 Syntactic congruences and algebras 

16 
2.3 The variety theorem 

21 



3 Positive varieties of tree languages 

29 



3.1 Ordered algebras 

30 
3.2 Positive variety theorem 

35 
3.3 Generalized positive variety theorem 

40 



4 Definability by monoids 

47 



4.1 Algebras definable by translation monoids 

49 
4.2 Tree languages definable by monoids 

52 
4.3 Definability by semigroups 

59 



5 Definability by ordered monoids 

65 



5.1 Ordered algebras vs. ordered monoids 

65 
5.2 Tree languages definable by ordered monoids 

69 
5.3 Examples of varieties 

76 



6 Tree algebras 

87 



6.1 Binary trees and tree algebras 

88 
6.2 Varieties of binary tree languages 

94 
6.3 Some algebraic properties of tree algebras 

100 



Epilogue 

107 






Index of Notation 

109 






References 

113 