arxiv: 2604.26723 · v1 · submitted 2026-04-29 · 🧮 math.AC

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On the binary relations defined using GD1 and 1GD inverses over infinite dimensional vector spaces

Diego Alba Alonso, Javier S\'anchez Gonz\'alez

Pith reviewed 2026-05-07 12:31 UTC · model grok-4.3

classification 🧮 math.AC

keywords GD1 inverse1GD inversebinary relationspartial ordersfinite potent endomorphismsAST decompositiongeneralized inversesinfinite dimensional vector spaces

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The pith

GD1 and 1GD binary relations define partial orders on finite potent endomorphisms over infinite-dimensional vector spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how binary relations defined by GD1 and 1GD generalized inverses can be characterized using the AST decomposition of finite potent endomorphisms. It proves that these relations form partial orders on the set of such endomorphisms, even when the underlying vector space is infinite-dimensional. A reader would care because this extends the known theory of generalized inverses from matrices to a broader setting without requiring finite dimension. The work provides algorithms for computing these inverses based on the decomposition.

Core claim

The GD1 and 1GD inverses of finite potent endomorphisms are characterized in terms of their AST decomposition, allowing the definition of binary relations that are shown to be partial orders on the set of all finite potent endomorphisms over arbitrary vector spaces, thereby completing the corresponding theory originally developed for matrices.

What carries the argument

The AST decomposition of a finite potent endomorphism, which decomposes it into parts that enable explicit description of the generalized inverses and the induced relations.

If this is right

Algorithms exist for computing GD1 and 1GD inverses using the AST decomposition.
The GD1 and 1GD relations are partial orders on finite potent endomorphisms.
This holds over infinite dimensional vector spaces.
The theory of these generalized inverses is completed for matrices by this extension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same decomposition technique might allow similar partial-order results for other generalized inverses beyond GD1 and 1GD.
The partial orders could induce natural chains or comparability classes among endomorphisms that might be studied for convergence properties in infinite dimensions.

Load-bearing premise

The AST decomposition exists and behaves consistently for all finite potent endomorphisms over any vector space, including infinite-dimensional ones.

What would settle it

A counterexample consisting of a finite potent endomorphism over an infinite-dimensional space where the GD1 relation fails to be transitive or reflexive would disprove the claims.

read the original abstract

The purpose of this article is to study certain binary relations of endomorphisms over infinite dimensional vector spaces defined by GD1 and 1GD generalized inverses. In order to do so, these generalized inverses are studied over arbitrary vector spaces (namely, infinite dimensional ones) using finite potent endomorphisms. We characterize them in terms of the AST decomposition of a finite potent endomorphism and we obtain algorithms for their respective computation. This theory is then used to characterize the GD1 and 1GD binary relations for finite potent endomorphisms in terms of the AST decomposition and to prove that they define partial orders in the set of finite potent endomorphisms, thus, completing the theory of these generalized inverses for matrices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves GD1 and 1GD relations form partial orders on finite-potent endomorphisms over any vector space by characterizing them with the AST decomposition.

read the letter

The main thing to know is that this paper completes the partial-order story for the GD1 and 1GD relations by extending them from matrices to finite-potent endomorphisms on arbitrary vector spaces and proving the order properties hold via the AST decomposition. It also supplies explicit characterizations and computation algorithms along the way. The work stays inside the finite-potent setting, where the eventual image is finite-dimensional, so the algebra stays manageable even when the ambient space is infinite-dimensional. That choice lets the authors adapt the matrix arguments without extra machinery. The proofs of reflexivity, antisymmetry, and transitivity follow from the block descriptions given by the decomposition. A minor point worth checking in the details is whether the invertibility on the image summand requires any extra algebraic condition in the infinite-dimensional case; the abstract presents the results as holding generally, but the derivations will show if any restrictions slipped in. This is narrow linear-algebra work aimed at people who already know the matrix versions of generalized inverses. Readers in that niche will get concrete characterizations and the order proofs they need to finish the theory. It is not broad enough to interest most operator theorists, but it fills a specific gap cleanly. I would send it for peer review. The claims are precise enough that referees can verify the decomposition steps and the order axioms without needing to guess at the authors' intent.

Referee Report

2 major / 2 minor

Summary. The paper studies binary relations on endomorphisms of infinite-dimensional vector spaces induced by GD1 and 1GD generalized inverses. Focusing on finite-potent endomorphisms, it characterizes these inverses and the induced relations via the AST decomposition V = im(T^∞) ⊕ ker(T^∞), supplies algorithms for computing the inverses, and proves that the GD1 and 1GD relations are partial orders on the set of all finite-potent endomorphisms, thereby extending the matrix theory to the infinite-dimensional setting.

Significance. If the central claims hold, the work completes the partial-order theory for GD1/1GD inverses by moving from matrices to finite-potent operators on arbitrary vector spaces. The AST-based characterizations and explicit algorithms would provide concrete tools for computation and verification in infinite dimensions, strengthening the algebraic framework for generalized inverses beyond finite-dimensional linear algebra.

major comments (2)

The manuscript's core results rest on the claim that every finite-potent endomorphism T admits an AST decomposition V = im(T^∞) ⊕ ker(T^∞) in which the restriction of T to im(T^∞) is invertible. This decomposition is invoked to characterize the GD1 and 1GD inverses and to verify reflexivity, antisymmetry, and transitivity of the induced relations. In infinite dimensions the eventual image is finite-dimensional, but the proof that T|im(T^∞) is invertible (and that the decomposition is functorial) is not supplied with the same detail as in the finite-dimensional case; the paper must either give a self-contained argument or state any additional hypotheses (e.g., algebraicity of T) required for the invertibility step. Without this, the subsequent partial-order proofs are not yet load-bearing.
The transitivity argument for the GD1 relation (presumably in the section following the AST characterization) translates the relation into block-matrix conditions on the AST components. The manuscript should verify that these block conditions remain sufficient when ker(T^∞) is infinite-dimensional; any implicit use of finite-dimensional rank-nullity or basis-extension arguments must be replaced by explicit infinite-dimensional reasoning.

minor comments (2)

The abstract promises 'algorithms for their respective computation,' yet the text presents only high-level descriptions; explicit pseudocode or step-by-step procedures would improve clarity.
Notation for the GD1 and 1GD relations and for the AST blocks should be introduced with a short table or diagram to aid readers unfamiliar with the prior matrix literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify areas where the infinite-dimensional arguments require additional explicit detail to be fully rigorous. We will revise the manuscript to supply self-contained proofs that avoid any reliance on finite-dimensional shortcuts.

read point-by-point responses

Referee: The manuscript's core results rest on the claim that every finite-potent endomorphism T admits an AST decomposition V = im(T^∞) ⊕ ker(T^∞) in which the restriction of T to im(T^∞) is invertible. This decomposition is invoked to characterize the GD1 and 1GD inverses and to verify reflexivity, antisymmetry, and transitivity of the induced relations. In infinite dimensions the eventual image is finite-dimensional, but the proof that T|im(T^∞) is invertible (and that the decomposition is functorial) is not supplied with the same detail as in the finite-dimensional case; the paper must either give a self-contained argument or state any additional hypotheses (e.g., algebraicity of T) required for the invertibility step. Without this, the subsequent partial-order proofs are not yet load-bearing.

Authors: We agree that the invertibility step merits a more explicit treatment in the infinite-dimensional setting. Finite potency of T means that im(T^k) stabilizes for large k and is finite-dimensional; by definition of the eventual image, T maps im(T^∞) onto itself and the kernel of the restriction is trivial, so the restriction is an automorphism. We will insert a self-contained lemma proving this directly from the definition of finite potency and the direct-sum decomposition, without invoking algebraicity or other extra hypotheses. The functoriality of the decomposition will also be verified explicitly. These additions will render the subsequent characterizations and partial-order proofs load-bearing. revision: yes
Referee: The transitivity argument for the GD1 relation (presumably in the section following the AST characterization) translates the relation into block-matrix conditions on the AST components. The manuscript should verify that these block conditions remain sufficient when ker(T^∞) is infinite-dimensional; any implicit use of finite-dimensional rank-nullity or basis-extension arguments must be replaced by explicit infinite-dimensional reasoning.

Authors: The block conditions arise from the operator equations T = T S T and the direct-sum decomposition; these are purely algebraic identities that hold irrespective of the dimension of ker(T^∞). No rank-nullity or basis-extension arguments appear in the proof. To address the concern explicitly, we will expand the transitivity section to restate each step using only the subspace inclusions and the definitions of the GD1 relation on the AST components, confirming that the argument remains valid when the kernel is infinite-dimensional. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent algebraic definitions and properties

full rationale

The paper defines GD1 and 1GD inverses over arbitrary vector spaces via finite potent endomorphisms, then characterizes the induced binary relations using the AST decomposition (V = im(T^∞) ⊕ ker(T^∞) with T invertible on the image summand). The partial-order proofs follow directly from verifying reflexivity, antisymmetry and transitivity on the AST blocks; these verifications use only the algebraic relations between the blocks and the definition of the inverses, without any fitted parameters, self-referential renaming, or load-bearing self-citation that collapses the claimed results back to the inputs. The extension to infinite dimensions is achieved precisely by restricting attention to finite-potent operators, for which the eventual image remains finite-dimensional and the decomposition is well-defined by standard linear-algebra arguments. No step reduces by construction to a prior result of the same authors or to a tautological re-expression of the starting assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence and properties of the AST decomposition for finite potent endomorphisms, together with the standard definition of GD1 and 1GD generalized inverses. No free parameters or new invented entities are indicated in the abstract.

axioms (2)

domain assumption Finite potent endomorphisms admit an AST decomposition that can be used to characterize generalized inverses.
Invoked to obtain algorithms and binary-relation characterizations.
standard math Standard properties of vector spaces and endomorphisms hold over arbitrary (including infinite-dimensional) fields.
Background assumption for working with endomorphisms.

pith-pipeline@v0.9.0 · 5420 in / 1395 out tokens · 45936 ms · 2026-05-07T12:31:26.442268+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 3 canonical work pages

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