ABSTRACT
This chapter provides a brief discussion of the abstract structure of vector lattices, which is the cornerstone of integration theory. A particularly important property of vector lattices is the distributivity. General lattices need not be distributive, which is somewhat surprising since the formulation of distributivity uses only order-theoretic concepts. However, the proof of distributivity in vector lattices uses their algebraic properties. The chapter discusses some basic properties of substructures of a vector lattice. The substructures include vector sublattice, solid subspace, and band. The chapter considers homomorphism and isomorphism of vector lattices and proves that an increasing linear map between two vector lattices need not be a homomorphism of vector lattices.
