Beyond True and False: Logic, Algebra, and Topology Florence, December 2014 Applications of the semisimple tensor product of MV-algebras

Joint work with S. Lapenta. Since the real interval [0, 1] is closed to the product operation, a natural problem was to find a complete axiomatization for the variety generated by the standard MV-algebra ([0, 1],⊕,¬, 0) endowed with the real product. If the product operation is defined as a bilinear function · : [0, 1] × [0, 1] → [0, 1] then the standard model is ([0, 1],⊕, ·,¬, 0); if the product is a scalar multiplication then the standard model is ([0, 1],⊕, {r | r ∈ [0, 1]},¬, 0), where the function x 7 → rx is a linear for any r ∈ [0, 1]. The approach based on the internal binary product led to the notion of PMV-algebra [1, 6, 7], while the approach based on the scalar multiplication led to the notion of Riesz MV-algebra [2]. In [4] we defined the fMV-algebras adding both an internal product and a scalar multiplication; in this case, our standard model is ([0, 1],⊕, ·, {r | r ∈ [0, 1]},¬, 0). The varieties of MV-algebras and Riesz MV-algebras are generated by their corresponding standard models; this is not the case for PMV-algebras and f MV-algebras: in these classes the standard models generate proper sub-varieties. There are obvious forgetful functors between the above mentioned classes of structures: MV PMV fMVRM