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arxiv: 2606.28477 · v1 · pith:T2I7EQA3new · submitted 2026-06-26 · 🧮 math.GR

On the Dominions of Certain Semigroups of Transformations

Halima H. Assiri , Jehan A. Albar (King Abdulaziz University) This is my paper

Pith reviewed 2026-06-30 01:27 UTC · model grok-4.3

classification 🧮 math.GR
keywords transformation semigroupsdominionorder-preserving mapsorder-decreasing mapsCatalan monoididempotent-generatedregular semigroups
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The pith

O_n is closed in T_n, and the dominions of D_n and C_n inside T_n are exactly the regular idempotent-generated subsemigroups with explicit size and idempotent counts.

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

The paper works inside the full transformation semigroup T_n on a finite chain X_n. It first proves that the subsemigroup O_n of all order-preserving maps is closed under composition. It then identifies the dominion of the order-decreasing subsemigroup D_n and the dominion of the Catalan monoid C_n inside T_n. Both dominions are shown to coincide with certain regular subsemigroups that are generated by their own idempotents, and the paper supplies formulas that count the elements and the idempotents of each dominion.

Core claim

In T_n the set O_n of order-preserving maps is a subsemigroup. The dominion Dom_{T_n}(D_n) equals a specific regular idempotent-generated subsemigroup of T_n whose elements and idempotents are counted by an explicit formula; the same holds for Dom_{T_n}(C_n), the dominion of the Catalan monoid of order-decreasing order-preserving maps.

What carries the argument

The dominion Dom_{T_n}(S) of a subsemigroup S inside T_n, characterized here via the order relations on the chain so that it equals a regular idempotent-generated subsemigroup.

If this is right

  • O_n being closed means composition of any two order-preserving maps remains order-preserving.
  • Both dominions are regular, so every element a satisfies a = a x a for some x inside the dominion.
  • The dominions are generated by their idempotents, so every element factors as a product of idempotent maps from the same set.
  • Explicit formulas give the exact number of elements and the exact number of idempotents in each dominion.

Where Pith is reading between the lines

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

  • The counting formulas may be checked against known sequences for small n to confirm the characterization.
  • The same order-based description of the dominion might extend to other subsemigroups defined by monotonicity conditions on chains.
  • Because the dominions are regular and idempotent-generated, their Green's relations or ideal structure can be read off from the idempotents alone.

Load-bearing premise

The order-decreasing and order-preserving conditions on the finite chain are enough to force the dominions to be precisely the regular idempotent-generated sets described.

What would settle it

For a concrete small n, exhibit an element of T_n that belongs to the dominion of D_n yet lies outside the regular idempotent-generated subsemigroup the paper identifies, or compute the stated counting formula for n=4 and compare against direct enumeration of the dominion.

read the original abstract

In the full transformation semigroup $T_n$ on a finite chain $X_n$, let $D_n=\{\alpha \in T_n:(\forall x \in X_n) \ x\alpha \leq x\}$ be the subsemigroup of all order-decreasing maps of $T_n$, and let $O_n=\{\alpha \in T_n:(\forall x ,y\in X_n) \ x \leq y \Rightarrow x\alpha \leq y\alpha\}$ be the subsemigroup of all order-preserving maps of $T_n$. The Catalan monoid $C_n$ is a semigroup of all order-decreasing and order-preserving full transformations of $X_n$. In this paper, it is shown that $O_n$ is closed in $T_n$. Also, the dominion of $D_n$ and the dominion of $C_n$ in $T_n$, denoted by $Dom_{T_n}(D_n)$ and $Dom_{T_n}(C_n)$, are characterized, and it is shown that they are regular idempotent-generated subsemigroups of $T_n$. Moreover, a formula for the number of their elements and their idempotents is given.

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

0 major / 3 minor

Summary. The paper studies subsemigroups of the full transformation semigroup T_n on the finite chain X_n. It defines D_n as the order-decreasing transformations, O_n as the order-preserving transformations, and C_n (the Catalan monoid) as those that are both. The claims are that O_n is closed in T_n, that the dominions Dom_{T_n}(D_n) and Dom_{T_n}(C_n) are regular idempotent-generated subsemigroups of T_n, and that explicit formulas exist for the cardinalities of these dominions and for the number of their idempotents.

Significance. If the characterizations and formulas hold, the work supplies concrete structural and enumerative information about dominions of order-related subsemigroups inside T_n, which is useful for the study of transformation semigroups and their dominions. The explicit cardinality formulas constitute a verifiable, falsifiable output.

minor comments (3)
  1. [Abstract] The abstract asserts the main results but does not outline the proof strategy or key lemmas; the introduction or §2 should supply a brief roadmap.
  2. [Introduction] Clarify the precise meaning of 'closed' for O_n (i.e., whether it means Dom_{T_n}(O_n) = O_n) and state the ambient semigroup explicitly when first introducing the dominion notation.
  3. Verify that the formulas for |Dom_{T_n}(D_n)| and the number of idempotents are accompanied by at least one small-n verification table (e.g., n=3 or n=4) to allow direct checking.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the paper, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines D_n, O_n and C_n directly from the order relations on the finite chain X_n, then proves O_n is closed in T_n and characterizes Dom_{T_n}(D_n) and Dom_{T_n}(C_n) as the regular idempotent-generated subsemigroups whose sizes and idempotent counts are given by explicit formulas. These steps rely on standard semigroup-theoretic arguments applied to the given definitions; no equation reduces to a fitted parameter, no result is obtained by renaming a known pattern, and no load-bearing premise depends on a self-citation whose content is itself unverified. The derivation is therefore self-contained against the external definitions of dominion and the order properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no free parameters, invented entities, or non-standard axioms are visible. Relies on background semigroup and order axioms.

axioms (1)
  • standard math Standard axioms of semigroup theory together with the definition of a finite chain and the order relations used to define D_n, O_n, and C_n.
    The paper invokes these to define the objects and state the closure and dominion properties.

pith-pipeline@v0.9.1-grok · 5750 in / 1264 out tokens · 32225 ms · 2026-06-30T01:27:19.389114+00:00 · methodology

discussion (0)

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Reference graph

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