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 ZA-filters on X. We observe that for σ-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.