We introduce semidirect and wreath products of finite ordered semigroups and extend some standard decomposition results to this case. 1 Introduction. All semigroups and monoids considered in this paper are either finite or free. The semidirect product is a powerful tool for studying finite semigroups, and it has been used in the literature to give structure theorems [8, 10, 15, 31] and classification theorems [1, 3, 4, 8, 32], especially in the context of th
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