Convex MV-algebras: many-valued logics meet decision theory (Q1615991)

In order to have logico-algebraic instruments in the framework of decision theory under uncertainty, the authors introduce the notion of convex combination in MV-algebras. In particular they define algebraic structures obtained by adding to MV-algebras a family of binary operations, indexed by real numbers \(\alpha \in [0,1]\), playing the role of convex combination of two points with parameter \(\alpha\). The obtained class of algebraic structures, called MV-algebras with convexity operators, are shown to be term equivalent to Riesz MV-algebras. Some results for such class are proved, as for example that no countable MV-algebra can be equipped with a family of convexity operators. Finally, MV-algebras with convexity operators are shown to be useful to approach the foundations of decision theory under uncertainty, in particula to give a logico-algebraic representation of the Anscombe-Aumann problem.