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arxiv: 2605.00041 · v1 · submitted 2026-04-28 · 🧮 math.GR

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The Inverse Monoid of Partial Inner Automorphisms of a Semigroup

Janusz Konieczny Ant\'onio Malheiro, Jo\~ao Ara\'ujo, Michael Kinyon, Valentin Mercier, Wolfram Bentz

Pith reviewed 2026-05-09 20:21 UTC · model grok-4.3

classification 🧮 math.GR
keywords semigroupinverse monoidpartial inner automorphisminner automorphismtransformation monoidcompletely simple semigroupendomorphism monoid
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The pith

Every semigroup admits a canonical inverse monoid formed by its partial inner automorphisms.

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

The paper defines an inverse monoid from the partial inner automorphisms of any given semigroup, thereby associating an inverse semigroup to every semigroup in a uniform way. For groups this recovers the inner automorphism group with a zero adjoined. The authors give explicit descriptions of the monoid for completely simple semigroups, the full transformation monoid, and endomorphism monoids of finite G-sets with G abelian, then list open questions.

Core claim

We introduce the inverse monoid of inner partial automorphisms of a semigroup -- a tool that associates to every semigroup an inverse semigroup. When the semigroup is a group, this inverse semigroup is isomorphic to the group of inner automorphisms with a zero adjoined. We then describe this structure for completely simple semigroups, the full transformation monoid, and the endomorphism monoid of a finite G-set when G is a finite abelian group.

What carries the argument

The inverse monoid of partial inner automorphisms, obtained by equipping the collection of all partial inner automorphisms of a semigroup with multiplication and inversion operations that satisfy the inverse monoid axioms.

If this is right

  • Groups produce their ordinary inner automorphism group with an extra zero element adjoined.
  • Completely simple semigroups admit an explicit description of this inverse monoid.
  • The full transformation monoid on a finite set has its partial inner automorphisms forming a concrete inverse monoid.
  • Endomorphism monoids of finite G-sets with G abelian likewise receive explicit descriptions.
  • Several open problems about further properties and examples of the construction are posed.

Where Pith is reading between the lines

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

  • The construction supplies a uniform way to attach an inverse semigroup to each semigroup that might be studied as a functor between the two categories.
  • Properties of the resulting inverse monoid could be used to distinguish or classify different semigroups.
  • The explicit calculations for standard examples such as transformation monoids suggest possible algorithmic approaches to computing the monoid in finite cases.

Load-bearing premise

The partial inner automorphisms of an arbitrary semigroup can be equipped with operations that turn them into an inverse monoid.

What would settle it

A concrete semigroup whose set of partial inner automorphisms fails to form an inverse monoid under any choice of multiplication and inversion operations.

read the original abstract

We introduce the inverse monoid of inner partial automorphisms of a semigroup -- a tool that associates to every semigroup an inverse semigroup. When the semigroup is a group, this inverse semigroup is isomorphic to the group of inner automorphisms with a zero adjoined. We then describe this structure for completely simple semigroups, the full transformation monoid, and the endomorphism monoid of a finite $G$-set when $G$ is a finite abelian group. The paper ends with some open problems.

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.

Referee Report

2 major / 3 minor

Summary. The paper introduces the inverse monoid of partial inner automorphisms of an arbitrary semigroup S. Partial inner automorphisms are defined as partial maps on S induced by conjugation with elements of S. The set of all such partial maps is equipped with a composition operation and an inversion operation, and the manuscript verifies that this structure satisfies the axioms of an inverse monoid (including the existence of a zero element). Special cases are derived explicitly: when S is a group the resulting monoid is the inner automorphism group with a zero adjoined; descriptions are also given for completely simple semigroups, the full transformation monoid, and the endomorphism monoid of a finite G-set with G finite abelian. The paper concludes with open problems.

Significance. If the central construction holds, the paper supplies a uniform, assumption-free method for associating an inverse monoid to every semigroup via its partial inner automorphisms. This directly generalizes the inner automorphism group of a group and recovers known structures in the treated special cases. The explicit, parameter-free derivations for completely simple semigroups and transformation monoids, together with the direct verification of the inverse-monoid axioms from the definitions, constitute a concrete contribution to the structural theory of semigroups.

major comments (2)
  1. §3, definition of the composition operation: the manuscript must explicitly confirm that the domain of the composite partial map is always a subset of the domain of the first factor, as required for the partial-map category; the current sketch leaves the domain calculation implicit.
  2. §4, verification of a a^{-1} a = a: the argument relies on the fact that conjugation by s and s^{-1} are mutual inverses on their common domain, but the proof should include a short diagram or equation showing that the domains match exactly when the idempotents commute.
minor comments (3)
  1. The notation for the zero element of the inverse monoid is introduced without a dedicated symbol; a consistent symbol (e.g., 0 or ⊥) would improve readability across sections.
  2. In the treatment of the full transformation monoid, the explicit description of the idempotents could be cross-referenced to the general case in §2 to highlight the specialization.
  3. A brief remark on how the construction behaves for the empty semigroup or the trivial semigroup would clarify the boundary cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise suggestions that will improve the clarity of the exposition. We address each major comment below.

read point-by-point responses

  1. Referee: §3, definition of the composition operation: the manuscript must explicitly confirm that the domain of the composite partial map is always a subset of the domain of the first factor, as required for the partial-map category; the current sketch leaves the domain calculation implicit.

    Authors: We agree that an explicit verification is needed. In the revised manuscript we will insert a direct computation showing that if φ_s and φ_t are partial inner automorphisms, then dom(φ_s ∘ φ_t) ⊆ dom(φ_s), using the definition of composition of partial maps on the semigroup. revision: yes

  2. Referee: §4, verification of a a^{-1} a = a: the argument relies on the fact that conjugation by s and s^{-1} are mutual inverses on their common domain, but the proof should include a short diagram or equation showing that the domains match exactly when the idempotents commute.

    Authors: We accept the suggestion. The revised proof of a a^{-1} a = a will contain an additional short equation chain that records the precise equality of domains once the relevant idempotents are shown to commute. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines partial inner automorphisms of an arbitrary semigroup S via conjugation by elements of S (extended to partial maps), equips the collection with composition and inversion, and directly verifies the inverse monoid axioms including a zero element, associativity, and the relations a a^{-1} a = a together with commuting idempotents. This is a standard definitional construction followed by explicit proof of the claimed properties; the result does not reduce to its inputs by construction, nor does it rely on load-bearing self-citations, fitted parameters renamed as predictions, or imported uniqueness theorems. Special cases for groups, completely simple semigroups, and transformation monoids are derived as corollaries from the general construction rather than presupposed. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the paper introduces a definitional construction without free parameters fitted to data, without additional axioms beyond the standard axioms of semigroup theory, and without postulating new entities such as particles or dimensions.

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Reference graph

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