arXiv subjects

Pronay Biswas

Publications and source records attributed to Pronay Biswas.

5 recordsLinked to original sources

arXiv2026

$M$-subgroups and normal-subgroup lattices of finite groups

Let $G$ be a group. A normal subgroup $E$ is an $M$-subgroup if $E\cap N_1\cap N_2\neq1$ whenever $N_1\cap N_2$, $E\cap N_1$, and $E\cap N_2$ are non-trivial for normal subgroups $N_1,N_2$. We characterise these subgroups as the essential or relatively uniform normal subgroups. For finite $G$, an $M$-subgroup is either trivial, contains exactly one minimal normal subgroup, or contains the socle. Consequently, every normal subgroup is an $M$-subgroup precisely when the socle has $G$-chief length at most two. We obtain an explicit $M$-closure operation and describe the minimal normal subgroups of finite direct products. For crown-based powers, we use the known comparability of normal subgroups with the socle to describe the normal lattice in terms of Boolean or subspace lattices and the quotient above the socle. We then determine when the $M$-lattice retains the normal lattice, classify normal quotients and their automorphism orbits, and characterise quotient maps compatible with $M$-closure. Finally, we compute the maximal length of chains of $M$-subgroups in terms of the chief length and socle length of $G$.

math.GR↗

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↗

arXiv2026

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↗

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↗

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↗