Our main issue was to understand the connection between Łukasiewicz logic with product and the Pierce-Birkhoff conjecture, and to express it in a mathematical way. To do this we define the class of f MV-algebras, which are MV-algebras endowed with both an internal binary product and a scalar product with scalars from [0, 1]. The proper quasi-variety generated by [0, 1], with both products interpreted as the real product, provides the desired framework: the normal form theorem of its corresponding logical system can be seen as a local version of the Pierce-Birkhoff conjecture.
In this paper we study the tensor product for MV-algebras, the algebraic structures of Łukasiewicz ∞-valued logic. Our main results are: the proof that the tensor product is preserved by the categorical equivalence between the MV-algebras and abelian lattice-order groups with strong unit and the proof of the scalar extension property for semisimple MValgebras. We explore consequences of this results for various classes of MV-algebras and lattice-ordered groups enriched with a product operation.
Abstract. In these notes we study the class of divisible MV-algebras inside the algebraic hierarchy of MV-algebras with product. We connect divisible MV-algebras with Q-vector lattices, we present the divisible hull as a categorical adjunction and we prove a duality between finitely presented algebras and rational polyhedra.
We study Łukasiewicz logic enriched with a scalar multiplication with scalars taken in [0, 1]. Its algebraic models, called Riesz MV-algebras, are, up to isomorphism, unit intervals of Riesz spaces with a strong unit endowed with an appropriate structure. When only rational scalars are considered, one gets the class of DMV-algebras and a corresponding logical system. Our research follows two objectives. The first one is to deepen the connections between functional analysis and the logic of Riesz MValgebras. The second one is to study the finitely presented MV-algebras, DMV-algebras and Riesz MV-algebras, connecting them from logical, algebraic and geometric perspective.
This paper introduces a logical analysis of convex combinations within the framework of Lukasiewicz real-valued logic. This provides a natural link between the fields of many-valued logics and decision theory under uncertainty, where the notion of convexity plays a central role. We set out to explore such a link by defining convex operators on MValgebras, which are the equivalent algebraic semantics of Lukasiewicz logic. This gives us a formal language to reason about the expected value of bounded random variables. As an illustration of the applicability of our framework we present a logical version of the Anscombe-Aumann representation result.
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