Trees and gaps from a construction scheme

We present natural constructions of trees and gaps using a quite general construction scheme. In particular, we solve a natural problem about $({\omega }_1,{\omega }_1)$-gaps. As it is well known $({\omega }_1,{\omega }_1)$-gaps can sometimes be filled in ${\omega }_1$-preserving forcing extensions of the set-theoretic universe. There are two natural conditions, dubbed $S$ and $T$ below, that guarantee the existence of such forcing extensions. The condition $T$ is a natural strengthening of the condition $S$ and was motivated by the numerous analogies between $({\omega }_1,{\omega }_1)$-gaps and certain trees of height ${\omega }_1\mathrm{.}$ It turns out that the condition $S$ is in fact equivalent to the existence of such forcing extensions but we show that the condition $T$ is strictly stronger by proving that it is consistent that there are fillable $({\omega }_1,{\omega }_1)$-gaps (i.e., S-gaps) but no T-gaps.