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Inclusion systems and amalgamated products of product systems

Here we generalize the concept of Skeide product, introduced by Skeide, of two product systems via a pair of normalized units. This new notion is called amalgamated product of product systems, and now the amalgamation can be done using contractive morphisms. Index of amalgamation product (when done through units) adds up for normalized units but for non-normalized units, the index is one more than the sum. We define inclusion systems and use it as a tool for index computations. It is expected that this notion will have other uses.