A long-standing problem asked whether every complemented subspace of a Banach lattice must be linearly isomorphic to a Banach lattice. This was recently answered negatively by De Hevia, Martínez-Cervantes, Salguero-Alarcón, and Tradacete, building on a construction of Plebanek and Salguero-Alarcón.

We will give an overview of the history of the problem, the main ideas behind the construction, and a modification which shows that there is a Banach lattice $E$ admitting a decomposition $E = X\oplus Y$ where neither $X$ nor $Y$ is isomorphic to a Banach lattice.

The talk is partially based on the author's article The class of Banach lattices is not primary, Forum Math. Sigma 14 (2026), e41.