Recently, MV-algebras with product have been investigated from different points of view. In particular a variety resulting from the combination of MV-algebras and product algebras has been introduced. The elements of this variety are called L$\Pi$-algebras. Even though the language of L$\Pi$-algebras is strong enough to describe the main properties of product and of Lukasiewicz 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. The resulting algebras, the L$\Pi_q$-algebras, are investigated in the present paper. In this paper we give a complete axiomatization of the quasivariety obtained in this way, and we show that such quasivariety is generated by the class of all L$\Pi_q$-algebras whose lattice reduct is the unit interval [0,1] with the usual order.
Continuous approximations of MV-algebras with product and product residuation