Decidability of Cancellative Extension of Monoidal T-norm Based Logic

It is known that the monoidal t-norm based logic (MTL) and many of its schematic extensions are decidable. The usual way how to prove decidability of some schematic extension of MTL is to show that the corresponding class of algebras of truth values has the finite model property (FMP) or the finite embeddability property (FEP). However this method does not work for the extensions whose corresponding classes of algebras have only trivial finite members. Typical examples of such extensions are the product logic and the cancellative extension of MTL (ΠMTL) because the only finite algebras belonging to the corresponding varieties are finite Boolean algebras. The product logic is known to be decidable because of its connection with ordered Abelian groups. However the decidability of ΠMTL was not known. This paper solves this problem.