bibkey: araujo2026completemappings authors: João Araújo; Wolfram Bentz; Peter J. Cameron; Kevin Hendrey; Michael Kinyon year: 2026 title: Complete Mappings of Semigroups doi: 10.48550/arXiv.2608.25092 url: https://arxiv.org/abs/2608.25092 claim: Problem 15.5 asks whether an orthodox semigroup with an ordering whose product is idempotent must have a complete mapping. strata_touched:

• D5/S0/Certificates/AraujoOrthodoxCompleteMappingRefutation license: citation-only triage: anchor

Complete mappings of semigroups

The abstract, printed page 1, defines the central notion:

A complete mapping of a semigroup S is a bijection α : S → S such that the map θ : S → S defined by xθ = x · xα is also a bijection.

Section 2, printed page 6, defines regularity:

An element a of a semigroup S is said to be regular if there exists b ∈ S such that aba = a. If every element of a semigroup is regular, then the semigroup itself is said to be regular.

Section 14, printed page 59, defines E-semigroups and orthodox semigroups:

A semigroup S is said to be an E-semigroup if the set E(S) of idempotents is a subsemigroup of S. A regular E-semigroup is said to be orthodox.

Theorem 14.3 on the same printed page states:

Theorem 14.3. Let S be a finite E-semigroup with a complete mapping. Then there exists an ordering c1, . . . , cn of all elements of S such that c1 · · · cn is an idempotent.

The following remark on printed page 59 identifies a zero-semigroup limitation:

We remark that for semigroups with zero this conclusion can be uninformative: nothing rules out the possibility that the product of elements given by Theorem 14.3 is zero, even if restricted to the non-zero elements.

Section 15, printed page 60, introduces the converse question:

The converse of Theorem 14.3 leads to the following question. An affirmative answer would generalize the Hall–Paige conjecture.

It then states:

Problem 15.5. Let S be an orthodox semigroup with an ordering c1, . . . , cn of all elements of S such that c1 · · · cn is an idempotent. Must S have a complete mapping?

The formal reading makes six choices. An ordering is a duplicate-free finite list containing every element. Its product is the left fold under semigroup multiplication. Idempotence means p * p = p. Orthodox means regular with the idempotents closed under multiplication. A complete mapping is a bijection α for which x ↦ x * α x is also bijective. The question is a universal claim over S : Type equipped with a semigroup structure.

The note records only Problem 15.5 and the definitions needed to state it. It does not make a claim about Problems 15.1–15.4 or 15.6–15.10, Theorem 4.4, Theorem 10.1, or the three-element semigroup obtained from C₂ by adjoining a zero.

Literature status

The source is arXiv:2608.25092v1, dated 2026-08-25; no journal version is listed. OpenAlex work W7204444461 reported zero citations, and Crossref returned no record. Peter Cameron’s 2026-08-27 blog announcement had no comments. Four MathDB searches found no entry for this problem. Semantic Scholar was not checked and remains ASSUMED-UNVERIFIED. These bounded searches do not establish exhaustive historical coverage or publication priority.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2608.25092
• URL: https://arxiv.org/abs/2608.25092
• Printed page 1: complete-mapping definition.
• Printed page 6: regularity definition.
• Printed page 59: E-semigroup and orthodox definitions, Theorem 14.3, and the zero-semigroup remark.
• Printed page 60: the lead-in and Problem 15.5.