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 a
fixed 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 the varieties of state-morphism algebras, in …