Vinay Madhusudanan | Searcharxiv

arXiv2026

On Complement and Supplement Ideals of Nearrings

In this article we study complement ideals, and the dual concept of supplement ideals, in nearrings, both of which are generalizations of the concept of complement in a bounded modular lattice. We prove fundamental properties of complements and supplements in arbitrary nearrings. We then establish Galois connections between the ideal lattices of a nearring and of its matrix nearrings, yielding one-to-one correspondences between their respective complement and supplement ideals. We also define graphs associated with complement and supplement ideals of nearrings and study some of their combinatorial properties such as girth and clique number.

math.RA↗

arXiv2026

Termination of the Lattice-Automorphism Tower for Direct Products of Symmetric Groups

Let $G$ be a finite group. Let $\mathcal{N}(G)$ be the lattice of normal subgroups ordered by inclusion, regarded as an abstract lattice. Define $\operatorname{LatAut}(G) := \operatorname{Aut}(\mathcal{N}(G))$. The \emph{LatAut tower} is the sequence defined by $G_0 = G$, $G_{n+1} = \operatorname{LatAut}(G_n)$. Let $G$ be a \emph{tower group} if $G \cong \prod_{k \geq 3} S_k^{a_k}$ with finitely many $a_k \neq 0$. We establish the following for tower groups. \emph{Product Formula.} $\operatorname{LatAut}\!\bigl(\prod_{k \geq 3} S_k^{a_k}\bigr) \cong S_{a_4} \times S_B$, where $B = \sum_{k \geq 3,\, k \neq 4} a_k$. \emph{Termination Theorem.} For every tower group $G_0$, we prove that $G_3 = 1$, and that this bound is sharp. The proof applies Goursat's lemma to classify $\mathcal{N}(G)$ into three families parameterised by admissible triples $(J,\mathbf{P},H)$ as sub-products, sign-parity elements, and mixed elements, and uses the Krull--Schmidt theorem to identify the direct factors $S_k^{(k,i)}$ as precisely the nontrivial indecomposable complemented elements of $\mathcal{N}(G)$ (the complemented elements being exactly the full sub-products). These results do not extend to groups outside the tower-group family.

math.GR↗