  We present a complete description of subdirectly irreducible state
BL-algebras as well as of subdirectly irreducible state-morphism BL-algebras.
In addition, we present a general theory of state-morphism algebras, that is,
algebras of general type with state-morphism which is an idempotent
endomorphism. We define a diagonal state-morphism algebra and we show that
every subdirectly irreducible state-morphism algebra can be embedded into a
diagonal one. We describe generators of varieties of state-morphism algebras,
in particular ones of state-morphism BL-algebras, state-morphism MTL-algebras,
state-morphism non-associative BL-algebras, and state-morphism pseudo
MV-algebras.
