Single chain completeness and some related properties

In 2010 Franco Montagna investigated two interesting properties of the axiomatic extensions of MTL, the single chain completeness (SCC) and the strong single chain completeness (SSCC). An axiomatic extension L of MTL enjoys the SCC if there is an L-chain A s.t. L is complete w.r.t. A, and L enjoys the SSCC if there is an L-chain A s.t. L is strongly complete w.r.t. A. Clearly the SSCC implies the SCC, whilst the converse implication has been left as an open problem. In this work we show that the SCC does not imply the SSCC, and that the SCC and SSCC are strongly related to some logical and algebraic properties relevant for substructural logics, as Halldén completeness (HC) and Deductive Maksimova variable separation property (DMVP). The HC will provide a logical characterization for the SCC, for every axiomatic extension of MTL, whilst the DMVP will be proved to be equivalent to the SSCC, for the n-contractive axiomatic extensions of BL. We conclude by studying the axiomatic extensions of MTL expanded with the Δ operator, by showing that SCC and SSCC always coincide, even in the first-order case.