Some of the basic substructural logics are shown by Ono to have the disjunction property (DP) by using cut elimination of sequent calculi for these logics. On the other hand, this syntactic method works only for a limited number of substructural logics. Here we show that Maksimova's criterion on the DP of superintuitionistic logics can be naturally extended to one on the DP of substructural logics over FL. By using this, we show the DP for some of the substructural logics (...) for which syntactic methods don't work well. (shrink)

This paper gives algebraic characterizations of Halld\'{e}n completeness, and of Maksimova's variable separation property and its deductive form. Though algebraic characterizations of these properties have been already studied for modal and superintuitionistic logics, e.g. in Wro\'{n}ski [12], pp.126--129), Maksimova [7], pp.168--184), [9], pp.99--112), a deeper analysis of these properties and non-trivial modifications of these results are needed to extend them to those for substructural logics, because of the lack of some structural rules in them. The first attempt in this direction (...) was made in the dissertation \cite{Kih06} of the first author. Results of this paper are partly announced also in Chapter 5 of the book [2]. (shrink)