Given two convex polytopes, the join, the cartesian product and the directsum of them are well understood. In this paper we extend these three kinds ofproducts to abstract polytopes and introduce a new product, called thetopological product, which also arises in a natural way. We show that these products have unique prime factorization theorems. We use this to compute the automorphism group of a product in terms of theautomorphism groups of the factors and show that (non trivial) products arealmost never regular or two-orbit polytopes. We finish the paper by studyingthe monodromy group of a product, show that such a group is always an extensionof a symmetric group, and give some examples in which this extension splits.
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