MV-algebras freely generated by finite Kleene algebras

If and are varieties of algebras such that any -algebra A has a reduct U(A) in , there is a forgetful functor that acts by on objects, and identically on homomorphisms. This functor U always has a left adjoint by general considerations. One calls F(B) the -algebra freely generated by the -algebra B. Two problems arise naturally in this broad setting. The description problem is to describe the structure of the -algebra F(B) as explicitly as possible in terms of the structure of the -algebra B. The recognition problem is to find conditions on the structure of a given -algebra A that are necessary and sufficient for the existence of a -algebra B such that . Building on and extending previous work on MV-algebras freely generated by finite distributive lattices, in this paper we provide solutions to the description and recognition problems in case is the variety of MV-algebras, is the variety of Kleene algebras, and B is finitely generated-equivalently, finite. The proofs rely heavily on the Davey-Werner natural duality for Kleene algebras, on the representation of finitely presented MV-algebras by compact rational polyhedra, and on the theory of bases of MV-algebras.