$(\odot, \oplus)$-Derivations and $(\ominus, \odot)$-Derivations on $MV$-algebras

In this paper, we introduce the notions of $(\odot, \oplus)$-derivations and $(\ominus, \odot)$-derivations for $MV$-algebras and discuss some related results. We study the connection between these derivations on an $MV$-algebra $A$ and the derivations on its boolean center. We characterize the isotone $(\odot, \oplus)$-derivations and prove that $(\ominus, \odot)$-derivations are isotone. Finally we determine the relationship between $(\odot, \oplus)$-derivation and $(\ominus, \odot)$-derivation for $MV$-algebras.