($\odot$, $\vee$)-derivations on MV-algebras

Let $A$ be an MV-algebra. An $(\odot ,\vee )$-derivation on $A$ is a map $d:A\to A$ satisfying: $d(x\odot y)=(d(x)\odot y)\vee (x\odot d(y))$ for all $x,y\in A$. This paper initiates the study of $(\odot ,\vee )$-derivations on MV-algebras. Several families of $(\odot ,\vee )$-derivations on an MV-algebra are explicitly constructed to give realizations of the underlying lattice of an MV-algebra as lattices of $(\odot ,\vee )$-derivations. Furthermore, $(\odot ,\vee )$-derivations on a finite MV-chain are enumerated and the underlying lattice is described.