2024/07/26 by Pronay Biswas, Biswas, Pronay, Sagarmoy Bag +3
Engineering · #Advanced Control Systems Optimization #FOS: Mathematics #Functional Analysis (math.FA) #Primary 54C40 #Rings and Algebras (math.RA) #Secondary 46E30
paper · pdf · doi:10.48550/arxiv.2408.10221
openalex publication_date 2024/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.