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If V and W are varieties of algebras such that any V-algebra A has a reductU(A) in W, there is a forgetful functor U: V->W that acts by A |-> U(A) onobjects, and identically on homomorphisms. This functor U always has a leftadjoint F: W->V by general considerations. One calls F(B) the V-algebra freelygenerated by the W-algebra B. Two problems arise naturally in this broadsetting. The description problem is to describe the structure of the V-algebraF(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 givenV-algebra A that are necessary and sufficient for the existence of a W-algebraB such that F(B) is isomorphic to A. Building on and extending previous work onMV-algebras freely generated by finite distributive lattices, in this paper weprovide solutions to the description and recognition problems in case V is thevariety of MV-algebras, W is the variety of Kleene algebras, and B is finitelygenerated--equivalently, finite. The proofs rely heavily on the Davey-Wernernatural duality for Kleene algebras, on the representation of finitelypresented MV-algebras by compact rational polyhedra, and on the theory of basesof MV-algebras.

MV-algebras freely generated by finite Kleene algebras