A systematic investigation to derive nonlinear lattice equations governed by partial difference equations (PΔΔE) admitting specific Lax representation is presented. Further it is shown that for a specific value of the parameter the derived nonlinear PΔΔE's can be transformed into a linear PΔΔE's under a global transformation. Also it is demonstrated how to derive higher order ordinary difference equations (OΔE) or mappings in general and linearizable ones in particular from the obtained nonlinear PΔΔE's through periodic reduction. The question of measure preserving property of the obtained OΔE's and the construction of more than one integrals (or invariants) of them is examined wherever possible.

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