MV-algebras freely generated by finite Kleene algebras

If V and W are varieties of algebras such that any V -algebra A has a reduct U(A) in W , there is a forgetful functor U:V→W that acts by A↦U(A) on objects, and identically on homomorphisms. This functor U always has a left adjoint F:W→V by general considerations. One calls F(B) the V -algebra freely generated by the W -algebra B. Two problems arise naturally in this broad setting. The description problem is to describe the structure of the V -algebra F(B) as explicitly as possible in terms of the structure of the W -algebra B. The recognition problem is to find conditions on the structure of a given V -algebra A that are necessary and sufficient for the existence of a W -algebra B such that F(B)≅A . 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 V is the variety of MV-algebras, W 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.