Injective Banach lattices: a survey

The work is aimed to survey recent results on injective Banach lattices, outline the Boolean-valued approach, and pose some open problems. The central idea of the investigation is the Boolean-valued transfer principle from $AL$-spaces to injective Banach lattices: Every injective Banach lattice is embedded into an appropriate Boolean-valued model, becoming an $AL$-space. To illustrate the method, a concrete description of injective Banach lattices similar to that of $AL$-spaces is presented and the isometric classification of injective Banach lattices is accomplished.