Concerning semirings of measurable functions

For a measurable space (X, A), let M^+ (X, A) be the commutative semiring of non-negative real-valued measurable functions with pointwise addition and pointwise multiplication. We show that there is a lattice isomorphism between the ideal lattice of M^+ (X, A) and the ideal lattice of its ring of differences M (X, A). Moreover, we infer that each ideal of M^+ (X, A) is a semiring z-ideal. We investigate the duality between cancellative congruences on M^+ (X, A) and Z₀-filters on X. We observe that for -algebras, compactness and pseudocompactness coincide, and we provide a new characterization for compact measurable spaces via algebraic properties of M^+ (X, A). It is shown that the space of (real) maximal congruences on M^+ (X, A) is homeomorphic to the space of (real) maximal ideals of the M (X, A). We solve the isomorphism problem for the semirings of the form M^+ (X, A) for compact and realcompact measurable spaces.