In the paper, we present a new theory of the non-disjoint serial decomposition. The non-disjoint decomposition is more general than the disjoint decomposition considered by researchers till now. Therefore, it produces often much better circuits. We also present our new decomposition tool DEMAIN. An original partition-based representation of Boolean functions and application of the non-disjoint serial decomposition are the main characteristics that distinguish our approach from another ones. Our total decomposition approach implemented in DEMAIN relies on: a partition-based representation of Boolean functions, an effective "balanced" decomposition strategy that switches between the parallel and the non-disjoint serial decomposition, and an appropriate graph coloring for a non-disjoint serial decomposition. Separately, some of these methods were already used in decomposition algorithms, however they were never applied together in a technology specific mapper targeted at a look-up table FPGA structure. In consequence, we applied the non-disjoint serial decomposition and parallel decomposition for efficient synthesis of FPGA-based circuits directed towards area or delay optimization. The experimental results demonstrate, that our decomposition tool DEMAIN performed better than all the other university and commercial tools available for comparison.