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.
"An Algebraic Approach to the Disjunction Property of Substructural Logics." Notre Dame J. Formal Logic 48 (4) 489 - 495, 2007. https://doi.org/10.1305/ndjfl/1193667706