arXiv subjects

Pronay Biswas

Publications and source records attributed to Pronay Biswas.

4 recordsLinked to original sources

arXiv2026

Maximal-Hull $z$-Ideals, Congruence Closures, and Coherent Frames of Commutative Semirings

We develop a spectral theory of $z$-ideals for commutative semirings. The lattice $\mathsf{ZId}(S)$ of $z$-ideals is a \emph{coherent frame} for every commutative semiring $S$ -- unconditionally, without cancellativity, subtractivity, or Noetherian hypothesis -- so the prime spectrum $\mathsf{Spec}_z(S)$ is spectral. Under an explicit finite-type hypothesis on the canonical congruence-generated closure~$g$, the lattice $\mathsf{Id}_{g}(S)$ of $g$-closed ideals is likewise a coherent frame, and $\mathsf{Spec}_g(S)$ is spectral and homeomorphic to the space of prime $g$-congruences. These frame results are accompanied by a regularity criterion: a semiring with all multiplicative idempotents complemented is von Neumann regular if and only if every principal ideal is a $z$-ideal, extending Mason's classical theorem from rings. Separating the maximal-ideal-hull $z$-closure from the maximal-congruence-hull $g$-closure -- operations that coincide in rings but diverge in semirings -- is a central theme, confirmed by explicit computations in $\mathbb{N}$ and power-set semirings. Both constructions carry a complete functorial formulation.

math.RA↗

arXiv2026

On the Subtractive Ideal Structure of Commutative Semirings

In the theory of commutative semirings, the lack of additive inverses creates a structural divergence between ideals and congruences that does not exist in ring theory. The aim of this article is to restore critical ideal-theoretic properties via the subtractive property. We first prove a subtractive analogue of Krull's existence theorem, guaranteeing the existence of $k$-prime ideals disjoint from multiplicative sets. We show that in arithmetic semirings, the distinction between $k$-irreducible and $k$-strongly irreducible ideals vanishes, a coherence that we show is preserved under localisation. We investigate the structural properties and coincidence phenomena among associated subclasses of $k$-ideals in Laskerian semirings, von Neumann regular semirings, unique factorisation semidomains, principal ideal semidomains, and weakly Noetherian semirings. Finally, within the framework of additively idempotent semirings, we tether subtractive ideal-theoretic structures to underlying order-theoretic constraints, thereby obtaining new characterizations of $k$-prime and $k$-semiprime ideals. In that process, we also establish that every absolutely $k$-prime ideal is $k$-prime and every $k$-maximal ideal is absolutely $k$-prime.

math.RA↗

arXiv2025

Radical-Ideal Functors, a Support Bifibration, and Quantale Completion for Commutative Semirings

We organize ordinary, subtractive ($k$-), and strong ideal theory of commutative semirings into a functorial framework. Radical extension is left adjoint to contraction and yields coherent-frame-valued functors naturally represented by the open-set frames of the corresponding prime spectra. The comparison from ordinary to $k$-radical ideals is a natural nucleus whose components are surjective and, under coherent Stone duality, correspond to dense sublocale embeddings. Ordinary, $k$-, and strong prime spectra form nested natural spectral functors, while universal support objects recover the spectra, radical frames, and complemented idempotents. Finite supports assemble into a Grothendieck bifibration with a canonical bicartesian section. For complete idealic semirings, $k$-ideal completion realizes a subtractive form of ideal quantale completion. We compute the induced monad, identify its restriction to frames with the classical ideal-lattice monad, and prove that its Eilenberg--Moore category is equivalent to the category of integral commutative quantales. Applications include a Stone-spectrum criterion for positive cones of $f$-rings and density criteria for $k$-prime spectra of $r$-semirings.

math.RA↗

arXiv2024

Concerning semirings of measurable functions

For a measurable space $(X,\mathcal{A})$, let $\mathcal{M}^+(X,\mathcal{A})$ 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 $\mathcal{M}^+(X,\mathcal{A})$ and the ideal lattice of its ring of differences $\mathcal{M}(X,\mathcal{A})$. Moreover, we infer that each ideal of $\mathcal{M}^+(X,\mathcal{A})$ is a semiring $z$-ideal. We investigate the duality between cancellative congruences on $\mathcal{M}^{+}(X,\mathcal{A})$ and $Z_{\mathcal{A}}$-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 $\mathcal{M}^+(X,\mathcal{A})$. It is shown that the space of (real) maximal congruences on $\mathcal{M}^+(X,\mathcal{A})$ is homeomorphic to the space of (real) maximal ideals of the $\mathcal{M}(X,\mathcal{A})$. We solve the isomorphism problem for the semirings of the form $\mathcal{M}^+(X,\mathcal{A})$ for compact and realcompact measurable spaces.

math.FA↗