arXiv subjects

Madhusudhana Rao Dasari

Publications and source records attributed to Madhusudhana Rao Dasari.

3 recordsLinked to original sources

arXiv2026

Universal Operator Envelopes and Independent Arithmetic Defects of Additive Ternary Gamma-Rings

For an additive ternary $Γ$-ring $T$, we construct the associative rng generated by the two-element, two-index operators on ternary modules and compare it with its realization on the regular module. For a finite-free two-step system $T_μ=E\oplus W$, two integral matrices control complementary parts of this comparison: the middle-relation matrix $B_μ$ presents the hidden quotient occurring in the integral regular kernel, while the visible flattening $Θ_μ$ controls the visible image, its saturation and its nonflat base-change correction. The torsion cokernels of these matrices define an ordered arithmetic defect pair; this pair records selected arithmetic data and is not asserted to be the whole regular kernel. The principal result is a constructive realization theorem: for every ordered pair $(G,H)$ of finite abelian groups, there is a torsion-free two-step ternary $\mathbb Z$-ring whose visible-saturation torsion is $G$ and whose middle-relation torsion is $H$. Thus the two torsion mechanisms, including their prime supports and invariant factors, can be prescribed independently. The proof builds two rank-$(2,1)$ atomic tensor families with complementary Smith profiles and establishes an orthogonal-sum theorem for their presentation matrices. We also identify the standard base-change obstruction attached to these two matrices, describe their determinantal loci, realize classical rank-one matrix-factorization periodicity in an example not induced by an associative $\mathbb Z$-bilinear binary product on its underlying group, and classify all $256$ integral $2\times2\times2$ sign tensors by the resulting defect data. All finite calculations use exact integer arithmetic and are independently reproducible.

math.RA↗

arXiv2026

Typed Congruences and Quotient-Critical Irreducibility in Complete Ternary $Γ$-Semirings

We study congruences and binary reducibility in completely additive ternary $Γ$-semirings through their two-sorted atomic cores. For an odd $m\geq3$, an abelian group $G$ of exponent dividing $m-1$, and $u,v\in G$, we construct a two-branch symmetric $m$-ary band $F_m(G;u,v)$. We prove \[ \Con(F_m(G;u,v))\cong\operatorname{Sub}(G)\times B_2 \] and classify every carrier--index congruence pair: $(θ,ϕ)$ is typed exactly when $θ$ is an ordinary congruence and $ϕ$ refines $θ$. We also give a quotient-by-quotient reducibility criterion. In the finite case, $F_m(G;u,v)$ is quotient-critically irreducible exactly when $G$ is a cyclic $p$-group and $u-v$ has order $p$. The specialization $G=\mathbb Z_2$ yields a four-point family $H_m$ extending the irreducible ternary example of Devillet and Mathonet. Its congruence lattice is $B_3$, its alternating core has exactly $36$ typed congruence pairs, and its automorphism group is $C_2$. Its seven proper quotients are reducible and form five isomorphism types; in arity five their exact reduction counts are determined. The powerset lift of $H_5$ is a $16$-element atom-total complete atomic Boolean ternary $Γ$-semiring with no atomic binary collapse, whereas each proper diagonal strong atom-saturated quotient has one. Its parity congruence is recovered on atoms by one explicitly specified depth-one polynomial inequation.

math.RA↗

arXiv2026

Cyclic crossed-product envelopes and arity-detecting relative $K$-theory of automorphism-derived $n$-ary algebras

Let $m=n-1$. For a unital associative algebra $A$, an automorphism $σ$, and $c\in A^\times$ satisfying $σ(c)=c$ and $σ^m=\operatorname{Ad}_c$, we study the Hosszú--Gluskin-type product $[a_0,\ldots,a_m]=a_0σ(a_1)\cdotsσ^{m-1}(a_{m-1})ca_m$. We prove that its endpoint-anchored all-slot envelope is the cyclic crossed product generated by $A\otimes A^{\mathrm{op}}$ and $u$, with $ux=σ^e(x)u$ and $u^m=c\otimes(c^{-1})^{\mathrm{op}}$. For the finite-order case $c=1$ and permutation algebras $A=k^X$, this gives an orbit--stabilizer block decomposition. A general dimension-defect theorem then computes the homotopy fibre of the forgetful $K$-theory map. In characteristic zero, transitive $C_m$-sets of every divisor length $d\mid m$ have the same ordinary spectrum $\K(k)^m$ but boundary $\mathbb Z/d$. The regular action gives $π_{-1}\cong\mathbb Z/m$, recovering the arity; over finite fields, the remaining torsion is governed by multiplicative orders.

math.RA↗