MV-algebras freely generated by finite Kleene algebras

TL;DR

This paper characterizes the structure and recognition conditions for free MV-algebras generated by finite Kleene algebras, extending previous work with duality and polyhedral methods.

Contribution

It provides explicit solutions to the description and recognition problems for free MV-algebras over finite Kleene algebras, using duality and geometric representations.

Findings

Explicit description of free MV-algebras generated by finite Kleene algebras.

Recognition criteria for MV-algebras to be generated by finite Kleene algebras.

Application of duality and polyhedral techniques to algebraic structure analysis.

Abstract

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.