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) is isomorphic to 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.Comment: 27 pages, 8 figures. Submitted to Algebra Universali

A characterization of MV-algebras free over finite distributive lattices

Mundici has recently established a characterization of free finitely generated MV-algebras similar in spirit to the representation of the free Boolean algebra with a countably infinite set of free generators as any Boolean algebra that is countable and atomless. No reference to universal properties is made in either theorem. Our main result is an extension of Mundici\u2019s theorem to the whole class of MV-algebras that are free over some finite distributive lattice