Gowers' Ramsey Theorem for generalized tetris operations

We prove a generalization of Gowers' theorem for FINk where, instead of the single tetris operation T:FINk→FINk−1, one considers all maps from FINk to FINj for 0≤j≤k arising from nondecreasing surjections f:{0,1,…,k}→{0,1,…,j}. This answers a question of BartoÅ¡ová and Kwiatkowska. We also describe how to prove a common generalization of such a result and the Galvin-Glazer-Hindman theorem on finite products, in the setting of layered partial semigroups introduced by Farah, Hindman, and McLeod.