## LPi Logic with Fixed Points

We study a system, $\mu$L$\Pi$, obtained by an expansion of L$\Pi$ logic with fixed points connectives. The first main result of the paper is that $\mu$L$\Pi$ is standard complete, i.e. complete with regard to the unit interval of real numbers endowed with a suitable structure.

We also prove that the class of algebras which forms algebraic semantics for this logic is generated, as a variety, by its linearly ordered members and that they are precisely the interval algebras of real closed fields. This correspondence is extended to a categorical equivalence between the whole category of those algebras and another category naturally arising from real closed fields.

Finally, we show that this logic enjoys implicative interpolation.

## Continuous approximations of MV-algebras with product and product residuation: a category-theoretic equivalence

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.