Continuous approximations of product implication in MV-algebras with product

Recently, MV-algebras with product have been investigated from different points of view. In particular, in [EGM01], a variety resulting from the combination of MV-algebras and product algebras (see [H98]) has been introduced. The elements of this variety are called ŁΠ-algebras. Even though the language of ŁΠ-algebras is strong enough to describe the main properties of product and of ukasiewicz connectives on [0,1], the discontinuity of product implication introduces some problems in the applications, because a small error in the data may cause a relevant error in the output. In this paper we try to overcome this difficulty, substituting the product implication by a continuous approximation of it. We investigate the resulting algebras, called ŁΠ_q -algebras, and we show that these algebras constitute a quasivariety which is generated by the class of all ŁΠ _q -algebras whose lattice reduct is the unit interval [0,1] with the usual order.