A new class of $MV$-algebras with product, called L$\Pi_q$-algebras, has been introduced. In these algebras, the discontinuous product residuation $\to_\pi$ is replaced by a continuous approximation of it. These algebras seem to be a good compromise between the need ofexpressiveness and the need of continuity of connectives. Following a good tradition in many-valued logic, in this paper we introduce a class of commutative $f$-rings with strong unit and with a sort of weak divisibility property, called $f$-quasifields, and we show that the categories of L$\Pi_q$-algebras and of $f$-quasifields are equivalent.