Gowers' Ramsey theorem for generalized tetris operations

Abstract: We prove a generalization of Gowers' theorem for $\mathrm{FIN}{k}$ where, instead of the single tetris operation $T:\mathrm{FIN}{k}\rightarrow \mathrm{FIN}{k-1}$, one considers all maps from $\mathrm{FIN}{k}$ to $\mathrm{FIN}_{j}$ for $0\leq j\leq k$ arising from nondecreasing surjections $f:\left{ 0,1,\ldots ,k+1\right} \rightarrow \left{ 0,1,\ldots ,j+1\right} $. This answers a question of Barto\v{s}ov\'{a} and Kwiatkowska. We also 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.