Advances in the theory of muLPi algebras

Recently an expansion of \(\rm L {\rm\Pi\frac{1}{2}}\) logic with fixed points has been considered. In the present work we study the algebraic semantics of this logic, namely \(\mu L\Pi\) algebras, from algebraic, model theoretic and computational standpoints.
We provide a characterisation of free \(\mu L\Pi\) algebras as a family of particular functions from \([0,1]^{n}\) to \([0,1]\).  We show that the first-order theory of linearly ordered \(\mu L\Pi\)-algebras enjoys quantifier elimination, being, more precisely, the model completion of the theory of linearly ordered \(\rm L {\rm\Pi\frac{1}{2}}\) algebras. Furthermore, we give a functional representation of any \(\rm L {\rm\Pi\frac{1}{2}}\) algebra in the style of Di Nola Theorem for MV-algebras and finally we prove that the equational theory of \(\mu L\Pi\) algebras is in PSPACE.

Advances in the theory of muLPi algebras

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.

LPi Logic with Fixed Points