  In [Mon11] are studied, for the axiomatic extensions of the monoidal t-norm
based logic ([EG01]), the properties of single chain completeness. On the other
side, in [GJKO07, Chapter 5] are studied many logical and algebraic properties
(like Halld\'en completeness, variable separation properties, amalgamation
property etc.), in the context of substructural logics. The aim of this paper
is twofold: first of all we will specialize the properties studied in [GJKO07,
Chapter 5] from the case of substructural logics to the one of extensions of
MTL, by obtaining some general characterization. Moreover we will show that
some of these properties are indeed strictly connected to the topics developed
in [Mon11]. This will help to have a better intuition concerning some open
problems of [Mon11].
