-ring representation of lattice homomorphisms on -spaces

In this paper, we study bounded lattice homomorphisms from [Formula: see text]-spaces over complete [Formula: see text]-finite measure spaces into Banach lattices. For [Formula: see text], we work on the [Formula: see text]-ring of finite-measure sets and on simple functions with finite-measure support. We show that such operators are determined by their values on indicator functions of finite-measure sets, which yields a component-valued set function satisfying a local Boolean property. The operator is recovered as the unique bounded extension of the associated integral on simple functions. We obtain an exact operator-norm formula in terms of a natural size functional of the induced set function and prove a converse construction. In the AL-space case, we derive a Radon–Nikodým description and an explicit norm identity. Endpoint variants for [Formula: see text] on finite measure spaces and for [Formula: see text] under a [Formula: see text]-order continuity hypothesis are also included.