MV-algebras freely generated by finite Kleene algebras

摘要

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