An arXiv preprint gives a sharp structural answer to a question about finite semigroups: any finite semigroup with a complete mapping must be regular. It also proves that a finite semigroup has such a mapping exactly when every principal factor, one of the structural pieces used in the analysis, has one.
A complete mapping is a bijection of a semigroup whose pointwise product map is also a bijection. Put simply, assigning a partner to each element and multiplying point by point must produce every element exactly once.
From structure to exact tests
The proofs combine Green-Rees structure theory with Hall-Paige and Burnside reductions, matching and flow arguments, total unimodularity, edge-colouring, incidence methods and estimates involving permanents.
One central family is the Rees matrix semigroup, a construction organized around a group, two index sets and a sandwich matrix. Under the paper's finite, normalized setting, a complete mapping exists exactly when at least one of four conditions holds: the group has odd order; its Sylow 2-subgroups, which capture its 2-power structure, are non-cyclic; the product of the two index-set sizes is even; or the sandwich matrix has an entry of even order.
When zero entries are permitted, the related Rees 0-matrix problem can be read from the matrix's zero-one pattern. For groups that have complete mappings, existence is equivalent to Hall-type inequalities for row and column neighborhoods, meaning conditions on whether each collection has enough nonzero neighbors, or, equivalently, to a balanced weighting of the nonzero positions.
The pattern test also applies when the underlying group has no complete mapping, provided at least one index set has even cardinality. In a separate odd-by-odd case, with both index sets odd, the paper proves nonexistence when the group has non-trivial cyclic Sylow 2-subgroups and a normalized matrix of odd-order entries has some entries replaced by zero.
The framework reaches several monoids
The classification reaches the full linear monoid, the set of all linear maps on a finite-dimensional vector space over a finite field of q elements. Every proper principal factor has a complete mapping, while the full monoid has one exactly when its general linear group does. The exceptions to existence are dimension 1 when q is odd and dimension 2 when q = 2.
For the partition monoid, every proper principal factor likewise has a complete mapping, and the full monoid has one exactly when the corresponding symmetric group does: n = 1 or n is at least 4. The full transformation monoid follows the same rule, with complete mappings exactly when n = 1 or n is at least 4.
A broad sufficient condition covers every finite aperiodic regular star-semigroup, and the paper names the planar partition, Motzkin and Jones monoids as consequences. For finite inverse semigroups, the criterion works class by class: in each J-class, either its maximal subgroups have complete mappings or the number of L-classes, equivalently R-classes, is even.
What remains outside the classification
These are exact theorems under stated hypotheses, not a single unrestricted answer for every finite case. The Rees results depend on conditions such as normalization, the group's 2-power structure, index-set parity and the pattern of zero entries, while the overall scope is finite semigroups and related finite structures.
The preprint leaves the infinite-semigroup analogue open, along with an unrestricted criterion for finite Rees 0-matrix semigroups and explicit or efficient constructions for every proper principal factor of the full linear monoid. The document is arXiv version 1, dated 25 Aug 2026.
Paper data and sources
Original title: Complete Mappings of Semigroups
Authors: João Araújo, Wolfram Bentz, Peter J. Cameron et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text