Preserving maximal monotonicity with applications in sum and composition rules

In this paper, we study maximal monotonicity preserving mappings on the Banach space X × X ^*. Indeed, for a maximal monotone set $M\subset <!--\; ?\; -->X\times <!--\; \times \; -->X^{\ast <!--\; *\; -->}$ and for a multifunction $T:X\times <!--\; \times \; -->X^{\ast <!--\; *\; -->}⊸<!--\; ?\; -->Y\times <!--\; \times \; -->Y^{\ast <!--\; *\; -->}$ , under some sufficient conditions on M and T we show that T(M) is maximal monotone. As two consequences of this result we get sum and composition rules for maximal monotone operators.