The Disjunction Property in the Class of Paraconsistent Extensions of Minimal Logic

We consider the disjunction property, $\mathbf{DP}$, in the class of extensions of minimal logic $\mathbf{L}_{j}$. Conditions are described under which $\mathbf{DP}$ is translated from the class $\mathbf{PAR}$ of properly paraconsistent extensions of the logics of class $\mathbf{L}_{j}$ into the class $\mathbf{INT}$ of intermediate extensions and the class $\mathbf{NEG}$ of negative extensions, and conditions for its being translated back into $\mathbf{PAR}$. The logic $\mathbf{L}_{F}$ in $\mathbf{PAR}$, which specifies conditions for $\mathbf{DP}$ to be translated from $\mathbf{PAR}$ into $\mathbf{NEG}$, is defined and is characterized in terms of $j$-algebras and Kripke frames. Moreover, we show that ${\mathbf L}_F$ is decidable and possesses the disjunction property.