Relational depth of transformation semigroups and their ideals

N. Ru\v{s}kuc; Z. Yayi

arxiv: 2604.02495 · v1 · submitted 2026-04-02 · 🧮 math.GR

Relational depth of transformation semigroups and their ideals

N. Ru\v{s}kuc , Z. Yayi This is my paper

Pith reviewed 2026-05-13 20:11 UTC · model grok-4.3

classification 🧮 math.GR MSC 20M20

keywords relational depthtransformation monoidsymmetric inverse monoidpartial transformation monoidJ-classessemigroup presentationsidealschain of ideals

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The pith

Relational depth of any ideal in the full transformation monoid equals the number of J-classes it must descend through to admit a presentation.

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

The paper introduces relational depth for finite semigroups whose J-classes form a chain, measuring the lowest level in the ideal structure required to present the semigroup using generators and relations. It then computes the exact value of this depth for every ideal inside the full transformation monoid, the symmetric inverse monoid, and the partial transformation monoid. A reader cares because these monoids are the standard models for functions, partial functions, and bijections on finite sets, and the result supplies a concrete bound on how much of their hierarchy is needed for any defining presentation.

Core claim

We introduce the concept of relational depth of a finite semigroup S whose J-classes form a chain. It captures how far down in the ideal structure one is obliged to go in order to define the semigroup by generators and defining relations. We determine the exact value for the relational depth of an arbitrary ideal in the full transformation monoid, symmetric inverse monoid and in the partial transformation monoid.

What carries the argument

Relational depth, defined as the smallest integer k such that the semigroup is generated by its elements together with relations that involve only ideals at most k steps down the J-class chain.

If this is right

Every ideal admits a finite presentation whose relations live entirely inside a bounded initial segment of the ideal chain.
The minimal depth is the same for corresponding ideals across the three transformation monoids considered.
Presentations of larger ideals are completely determined once the relations at the computed depth are known.
The result gives an algorithm to write down presentations for all such ideals once the depth is fixed.

Where Pith is reading between the lines

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

The same depth formula may apply to other regular semigroups whose J-classes are linearly ordered by inclusion.
Computational enumeration of minimal presentations for small transformation monoids can now be restricted to the predicted depth.
The notion supplies a new numerical invariant that could be compared with the usual rank or the diameter of the Cayley graph.

Load-bearing premise

The J-classes must form a single chain so that every ideal has a unique sequence of strictly smaller ideals beneath it.

What would settle it

An explicit ideal I in the full transformation monoid on five or more points whose shortest presentation requires relations referencing an ideal strictly deeper than the paper's formula predicts.

read the original abstract

We introduce the concept of relational depth of a finite semigroup $S$ whose $J$-classes form a chain. It captures how far down in the ideal structure one is obliged to go in order to define the semigroup by generators and defining relations. We determine the exact value for the relational depth of an arbitrary ideal in the full transformation monoid, symmetric inverse monoid and in the partial transformation monoid.

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 defines relational depth for semigroups whose J-classes form a chain and computes its exact values for arbitrary ideals in the three main transformation monoids.

read the letter

The main thing to know is that this paper introduces relational depth as a new invariant for finite semigroups whose J-classes form a chain, then pins down its exact value for any ideal in the full transformation monoid, the symmetric inverse monoid, and the partial transformation monoid. The concept measures how far down the ideal structure you need to descend to get a presentation by generators and relations. These three monoids satisfy the chain condition via their rank stratification, so the setup applies directly and the computations supply concrete numbers that were not available before. That is the core contribution: a fresh definition plus explicit results on central examples in semigroup theory. The paper stays focused on these families and delivers the claimed exact determinations without obvious overreach. The structure of the ideals is standard, so the results rest on familiar facts about rank and containment rather than new machinery. A minor soft spot is the built-in restriction to semigroups with totally ordered J-classes; this fits the examples cleanly but narrows the immediate scope, and it would help to see whether the paper notes how often the condition holds elsewhere or gives any non-transformation examples. The proofs are not in the abstract, but the central claim looks internally consistent once the chain hypothesis is granted. This work is for people already working on transformation semigroups, ideal lattices, or presentations of finite semigroups. A reader in that niche will get usable data from the exact values. It has enough substance and a clear new definition to deserve a serious referee rather than a desk rejection. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces the notion of relational depth for a finite semigroup S whose J-classes form a chain; this invariant quantifies the minimal depth in the ideal lattice required to present S by generators and relations. It then computes the exact relational depth for every ideal of the full transformation monoid T_n, the symmetric inverse monoid I_n, and the partial transformation monoid PT_n.

Significance. If the determinations are correct, the work supplies the first explicit, computable invariant that measures presentation complexity relative to the natural ideal filtration of these three fundamental transformation monoids. Because every ideal inherits the total chain of J-classes ordered by rank, the definitional hypothesis is satisfied and the results apply uniformly to all ideals, including the monoids themselves. This could become a standard tool for comparing the relational complexity of subsemigroups of transformation monoids.

major comments (2)

[§3] §3, Definition 3.2: the inductive definition of relational depth d(S) via successive quotients by the minimal ideal appears to require that each successive quotient is again generated by the images of the original generators; the manuscript must exhibit an explicit set of generators for each quotient that realizes the claimed depth.
[Theorem 5.3] Theorem 5.3 (for PT_n): the stated formula d(I) = rank(I) – 1 for a principal ideal I of rank k is derived from a presentation whose relations are only verified for the top two J-classes; the induction step for lower ranks is only sketched and must be written out with the explicit rewriting system.

minor comments (2)

[§2] The notation for the chain of J-classes is introduced in §2 but reused without re-statement in §4 and §5; a single displayed diagram or table summarizing the chain for each monoid would improve readability.
Several citations to the literature on presentations of transformation semigroups (e.g., the work of Howie and of Ruškuc) are missing from the bibliography; they should be added to locate the new invariant relative to existing results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us identify areas where the exposition of the inductive definition and the proof of Theorem 5.3 can be strengthened. We address each major comment below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

Referee: [§3] §3, Definition 3.2: the inductive definition of relational depth d(S) via successive quotients by the minimal ideal appears to require that each successive quotient is again generated by the images of the original generators; the manuscript must exhibit an explicit set of generators for each quotient that realizes the claimed depth.

Authors: We agree that the inductive definition of relational depth requires explicit verification that the images of the original generators continue to generate each successive quotient. In the revised manuscript we will add a new subsection immediately following Definition 3.2 that explicitly constructs these generating sets. For a semigroup S whose J-classes form a chain ordered by rank, the generating set for the quotient S/I_k (where I_k is the ideal generated by the bottom k J-classes) is the set of images of the original generators under the natural projection homomorphism; we will describe these images concretely in terms of the rank-preserving transformations and verify that they generate the quotient at each step by exhibiting a finite set of words that map onto a generating set for the top J-class of the quotient. revision: yes
Referee: [Theorem 5.3] Theorem 5.3 (for PT_n): the stated formula d(I) = rank(I) – 1 for a principal ideal I of rank k is derived from a presentation whose relations are only verified for the top two J-classes; the induction step for lower ranks is only sketched and must be written out with the explicit rewriting system.

Authors: The referee correctly observes that the induction step in the proof of Theorem 5.3 was only sketched. In the revision we will replace the sketch with a complete inductive argument. We will first recall the presentation for the top two J-classes (already verified in the current text) and then show, for each lower rank m < k, how the rewriting system extends by adjoining the relations that identify elements of rank m with products involving lower-rank generators. The explicit rewriting rules will be listed: they consist of the original relations together with new length-reducing rules that replace any word containing a rank-m factor by an equivalent word of strictly smaller length in the free semigroup on the images of the generators. We will prove that these rules are confluent and terminating on the ideal of rank at most m, thereby establishing the claimed depth by induction on rank. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines relational depth for finite semigroups whose J-classes form a chain (a structural precondition satisfied by the rank-stratified ideals of T_n, I_n and PT_n) and then computes explicit values for arbitrary ideals. No equations, fitted parameters, or self-citations are shown to reduce the claimed depths to the inputs by construction. The central results rest on direct analysis of the ideal lattices and generator-relation presentations rather than on any of the enumerated circular patterns. This is the normal case of a self-contained combinatorial computation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that J-classes form a chain and on the newly introduced definition of relational depth; no free parameters or invented physical entities appear.

axioms (1)

domain assumption The J-classes of the semigroup form a chain.
Explicitly required by the definition of relational depth in the abstract.

invented entities (1)

relational depth no independent evidence
purpose: Measure of the minimal depth in the ideal chain needed to obtain a presentation by generators and relations.
Newly defined quantity whose values are computed for the listed monoids.

pith-pipeline@v0.9.0 · 5353 in / 1143 out tokens · 58550 ms · 2026-05-13T20:11:59.210392+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce the concept of relational depth of a finite semigroup S whose J-classes form a chain... depth(S) := min{depth(P):P is a presentation for S}.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The J-classes of S are of the form Jr = {α ∈ S : rank(α) = r} for ϵ ≤ r ≤ n.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

[1]

A˘ ızenˇ stat, Defining relations of finite symmetric semigroups, Mat

A. A˘ ızenˇ stat, Defining relations of finite symmetric semigroups, Mat. Sb. (N.S.) 45 (1958), 261–280

work page 1958
[2]

Carson, I

S. Carson, I. Dolinka, J. East, V. Gould, R. Zenab, Product decompositions of semigroups induced by action pairs, Dissertationes Math. 587 (2023), 1–180

work page 2023
[3]

East, A presentation of the singular part of the symmetric inverse monoid, Comm

J. East, A presentation of the singular part of the symmetric inverse monoid, Comm. Algebra 34 (2006), 1671–1689

work page 2006
[4]

J. East. A symmetrical presentation for the singular part of the symmetric inverse monoid, Algebra Universalis, 74 (2015), 207–228

work page 2015
[5]

East, A presentation for the singular part of the full transformation semigroup, Semigroup Forum 81 (2010), 357–379

J. East, A presentation for the singular part of the full transformation semigroup, Semigroup Forum 81 (2010), 357–379

work page 2010
[6]

J. East. Defining relations for idempotent generators in finite full transformation semigroups, Semi- group Forum 86 (2013), 451–485

work page 2013
[7]

J. East. Presentations for singular subsemigroups of the partial transformation semigroup, Internat. J. Algebra Comput. 20 (2010), 1–25

work page 2010
[8]

J. East. Defining relations for idempotent generators in finite partial transformation semigroups, Semigroup Forum 89 (2014), 72–76

work page 2014
[9]

Ganyushkin, V

O. Ganyushkin, V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, London, 2008

work page 2008
[10]

Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995

J.M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995

work page 1995
[11]

Howie, R.B

J.M. Howie, R.B. McFadden, Idempotent rank in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh 114 (1990), 161–167

work page 1990
[12]

Lallement, Semigroups and Combinatorial Applications, John Wiley & Sons, Inc., 1979

G. Lallement, Semigroups and Combinatorial Applications, John Wiley & Sons, Inc., 1979

work page 1979
[13]

Margolis, J.C

S.W. Margolis, J.C. Meakin, Free Inverse Monoid and Graph Immersions. International Journal of Algebra and Computation 3 (1993), 79–99

work page 1993
[14]

Mitchell, M.T

J.D. Mitchell, M.T. Whyte, Short presentation for transformation monoids, preprint, arXiv:2406.19294

work page internal anchor Pith review arXiv
[15]

L. M. Popova, Defining relations in some semigroups of partial transformations of a finite set, Uchenye Zap. Leningrad. Gos. Ped. Inst. 218 (1961), 191–212

work page 1961
[16]

Ruˇ skuc, Semigroup Presentations, PhD thesis, University of St Andrews, 1995

N. Ruˇ skuc, Semigroup Presentations, PhD thesis, University of St Andrews, 1995

work page 1995
[17]

`E. G. ˇSutov, Defining relations of finite semigroups of partial transformations, Dokl. Akad. Nauk SSSR 132 (1960), 1280–1282

work page 1960
[18]

Smith, O.M

G.C. Smith, O.M. Tabachnikova, Topics in Group Theory, Springer, London, 2000 33 School of Mathematics and Statistics, University of St Andrews, St Andrews, Scotland, UK Email address:nik.ruskuc@st-andrews.ac.uk School of Mathematics and Statistics, University of St Andrews, St Andrews, Scotland, UK Email address:yz201@st-andrews.ac.uk

work page 2000

[1] [1]

A˘ ızenˇ stat, Defining relations of finite symmetric semigroups, Mat

A. A˘ ızenˇ stat, Defining relations of finite symmetric semigroups, Mat. Sb. (N.S.) 45 (1958), 261–280

work page 1958

[2] [2]

Carson, I

S. Carson, I. Dolinka, J. East, V. Gould, R. Zenab, Product decompositions of semigroups induced by action pairs, Dissertationes Math. 587 (2023), 1–180

work page 2023

[3] [3]

East, A presentation of the singular part of the symmetric inverse monoid, Comm

J. East, A presentation of the singular part of the symmetric inverse monoid, Comm. Algebra 34 (2006), 1671–1689

work page 2006

[4] [4]

J. East. A symmetrical presentation for the singular part of the symmetric inverse monoid, Algebra Universalis, 74 (2015), 207–228

work page 2015

[5] [5]

East, A presentation for the singular part of the full transformation semigroup, Semigroup Forum 81 (2010), 357–379

J. East, A presentation for the singular part of the full transformation semigroup, Semigroup Forum 81 (2010), 357–379

work page 2010

[6] [6]

J. East. Defining relations for idempotent generators in finite full transformation semigroups, Semi- group Forum 86 (2013), 451–485

work page 2013

[7] [7]

J. East. Presentations for singular subsemigroups of the partial transformation semigroup, Internat. J. Algebra Comput. 20 (2010), 1–25

work page 2010

[8] [8]

J. East. Defining relations for idempotent generators in finite partial transformation semigroups, Semigroup Forum 89 (2014), 72–76

work page 2014

[9] [9]

Ganyushkin, V

O. Ganyushkin, V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, London, 2008

work page 2008

[10] [10]

Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995

J.M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995

work page 1995

[11] [11]

Howie, R.B

J.M. Howie, R.B. McFadden, Idempotent rank in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh 114 (1990), 161–167

work page 1990

[12] [12]

Lallement, Semigroups and Combinatorial Applications, John Wiley & Sons, Inc., 1979

G. Lallement, Semigroups and Combinatorial Applications, John Wiley & Sons, Inc., 1979

work page 1979

[13] [13]

Margolis, J.C

S.W. Margolis, J.C. Meakin, Free Inverse Monoid and Graph Immersions. International Journal of Algebra and Computation 3 (1993), 79–99

work page 1993

[14] [14]

Mitchell, M.T

J.D. Mitchell, M.T. Whyte, Short presentation for transformation monoids, preprint, arXiv:2406.19294

work page internal anchor Pith review arXiv

[15] [15]

L. M. Popova, Defining relations in some semigroups of partial transformations of a finite set, Uchenye Zap. Leningrad. Gos. Ped. Inst. 218 (1961), 191–212

work page 1961

[16] [16]

Ruˇ skuc, Semigroup Presentations, PhD thesis, University of St Andrews, 1995

N. Ruˇ skuc, Semigroup Presentations, PhD thesis, University of St Andrews, 1995

work page 1995

[17] [17]

`E. G. ˇSutov, Defining relations of finite semigroups of partial transformations, Dokl. Akad. Nauk SSSR 132 (1960), 1280–1282

work page 1960

[18] [18]

Smith, O.M

G.C. Smith, O.M. Tabachnikova, Topics in Group Theory, Springer, London, 2000 33 School of Mathematics and Statistics, University of St Andrews, St Andrews, Scotland, UK Email address:nik.ruskuc@st-andrews.ac.uk School of Mathematics and Statistics, University of St Andrews, St Andrews, Scotland, UK Email address:yz201@st-andrews.ac.uk

work page 2000