2023/11/07 by Soichiro Fujii, Fujii, Soichiro · 1 citation
Computer Science · Decision Sciences · Mathematics · #06B10 #06D05 #06D22 #06F07 #06F25 #13A15 #16Y60 #18F70 #18F75 #Advanced Algebra and Logic #Category Theory (math.CT) #Commutative Algebra (math.AC) #FOS: Mathematics #Fuzzy and Soft Set Theory #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2311.03862
openalex publication_date 2023/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An ordered semiring is a commutative semiring equipped with a compatible preorder. Ordered semirings generalise both distributive lattices and commutative rings, and provide a convenient framework to unify certain aspects of lattice theory and ring theory. The ideals of an ordered semiring A form a commutative integral quantale Idl(A), and similarly, the radical ideals of A form a (spatial) frame Rad(A). We characterise Idl and Rad as the left adjoints of the (non-full) inclusion functors from the categories of commutative integral quantales and of frames, respectively, to that of ordered semirings and subadditive morphisms between them. The (sober) topological space pt(Rad(A)) corresponding to Rad(A) is homeomorphic to the space Spec(A) of prime ideals of A.