Sharp arity bounds for clonoids from mixed-exponent groups
September 9, 2026
Abstract
Let be a prime and . We prove that every clonoid from to a module on which is invertible is generated by its -ary functions. The stronger uniform reconstruction identity holds over , and the bound is optimal in general, even for finite field targets. In particular, this answers the mixed-exponent case singled out by Fioravanti, Kompatscher and Rossi. The proof combines Kovács’s theorem on finite-field matrix semigroup algebras with the character decomposition of a principal congruence subgroup. A correction on the -torsion submodule handles the zero character. A Fourier functional gives the matching lower bound for actual clonoid generation.
1. Introduction and main result
For algebras and , a clonoid from to is a collection of functions , for positive integers , closed under precomposition by term operations of and postcomposition by term operations of . When both algebras are modules, these operations are linear combinations over their respective scalar rings. We regard an abelian group as a module over the integers.
Mayr and Wynne [2] proved finiteness of the clonoid lattice for coprime finite modules when the source has a distributive submodule lattice. Fioravanti, Kompatscher and Rossi [1] established sharp uniform arity bounds for finite vector-space sources. Their Conjecture 1 predicts finiteness for all pairs of finite modules of coprime order. Their stronger Conjecture 59 asks for uniform reconstruction for finite abelian -groups, and Question 60 asks specifically about .
We settle that question and the family obtained by adjoining any number of elementary abelian factors. The case below recovers a known cyclic case of [2]; the mixed-exponent cases have .
For a matrix over a ring , its inner rank is the least for which with having columns and having rows. The zero matrix has inner rank zero. A substitution with inner rank at most factors through variables.
Theorem 1.1.
Let be a prime, , and
For every there are coefficients , indexed by matrices of inner rank at most , such that, for every -module and every function ,
Consequently every clonoid from to a module whose scalar ring contains is generated by its -ary part.
For every there is a finite field of characteristic different from and a clonoid from to the regular -module that is not generated by its -ary part.
The matrix in (1) acts by integer scalar combinations on the whole group . Its actions on the cyclic and elementary factors are compatible reductions of the same matrix. The proof does not assume that one can choose these actions independently.
Corollary 1.2.
If is a finite module, there are finitely many clonoids from to if and only if does not divide .
Proof.
If , multiplication by has an inverse given by an integer scalar on . Thus (1) uses term operations of . Each clonoid is determined by its -ary part, a subset of the finite set . Conversely, when , the common-prime-divisor result of Mayr and Wynne [2, Theorem 1.3] gives an infinite ascending chain of clonoids.
2. Matrix operators and the finite-field input
If is a finite set, write for the free module with basis , , over a commutative ring . A map induces the pushforward . Transposing an identity between these operators gives the corresponding identity for precomposition on functions.
We use the following consequence of Kovács’s theorem [3]. It is also the operator form of the finite-vector-space reconstruction theorem in [1, Theorem 49]. We include a derivation from the older matrix-algebra result.
Lemma 2.1.
Let be a finite field of characteristic , and let be a commutative ring in which is invertible. For integers , the identity on is a -linear combination of operators
Proof.
If , the set has one element and the zero matrix suffices. If , use the identity matrix. For , Kovács’s theorem says that the span of singular matrices in has its own multiplicative identity ; see also [4, Theorem 2.1]. For each choose a singular projection whose image contains the column space of . Since , its action gives
Thus acts as the identity. Expand in singular matrices. Factor each such matrix as through , where . By induction, insert a rank-at-most- operator identity on between the maps induced by and . Each resulting matrix has the form and rank at most . This proves the assertion by induction on .
We now fix and put
Here is the cyclotomic extension of with basis . Let
Write its basis elements as , where the columns of record the elementary coordinates. For define
A bar denotes reduction modulo . Let be the image of in , and set
This is a two-sided ideal of , since multiplication of matrices cannot increase inner rank. Our task is to show .
Whenever a field matrix of rank at most is lifted to , we choose a factorization through , lift entrywise, and use . This produces a lift of inner rank at most . An arbitrary entrywise lift of need not have that property.
3. The nonzero congruence characters
The principal congruence subgroup
is elementary abelian. The expression is independent of the chosen lifts of the entries of . For put
Orthogonality of the characters of the additive matrix group gives
These identities hold over : all normalizing denominators are powers of , and the sum of a nontrivial additive character is zero.
Lemma 3.1.
If , then on . If has rank one, then .
Proof.
Fix a residue vector and an elementary matrix . If , then fixes every with this residue. If , its orbit consists of
and the stabilizer is . The annihilator of this stabilizer under the trace pairing consists precisely of matrices , : indeed , and both spaces have dimension . Therefore only the zero character and rank-one characters occur.
We can assume , since otherwise every already belongs to . A nonzero rank-one matrix is conjugate to one of
For the first case use its nonzero image eigenvector and its kernel; for the second use a vector in the image and a preimage of that vector. A conjugating matrix over lifts to . Conjugating (3) correspondingly conjugates and preserves , so it is enough to handle these forms.
Set and in the first case, and and in the second. For and , use the integer representative of in and define
These vectors form a basis of . To see this, the only possible nonzero residues are , and each such orbit contributes the one-dimensional character summand displayed in (5). In particular,
Take and a factor-preserving lift as above. Put
It fixes exactly and satisfies . Expanding (5) and then using (6) gives
Indeed, the summand indexed by acquires the phase , which cancels its coefficient phase. All summands then agree. This calculation does not assume that is invertible.
Apply Lemma 2.1 to the last rows of , which have columns, while keeping its first rows and fixed. If are the resulting coefficients with , then (7) yields the operator identity
Each has inner rank at most . Since is a two-sided ideal, the right hand side belongs to it. This proves the lemma, including by the singleton case of Lemma 2.1.
4. The zero character and coefficient descent
Let be the span of the basis elements with . This is an -submodule. By writing , we identify it with
on which acts through .
The image contains . Modulo it has basis
where the choice of the lift is immaterial. For any , averaging the output orbit shows
For the first case, sends every output basis element to the average of all lifts of the same nonzero residue, so multiplicities sum to one. This also removes the carry in . For the second case, all output basis elements lie in .
Thus the compressed action in (11) is the field action on modulo the invariant subspace in which the first column is zero. Choose a field identity from Lemma 2.1, with coefficients and , and factor-preserving lifts . Define
Here , and by Lemma 3.1. Equation (9) says that is the identity on . Equations (4) and (11) say that . Consequently
This proves the desired operator identity over .
All matrices of the operators in the basis have integer entries. Apply to their coefficients the -linear map
It fixes and therefore descends the identity to . No averaging over a group of order is involved. Transposing now gives (1) for -valued functions, and the same finite coefficient identity holds for functions into every -module.
To obtain clonoid generation, write each matrix in (1) as through variables. If belongs to a clonoid , then belongs to its -ary part, and is generated by that part. The scalar coefficients are term operations on the target module. This proves the upper bound in Theorem 1.1.
Remark 4.1.
The zero-character image need not be stable under singular substitutions. The proof uses the sandwiches and the invariant submodule explicitly. Replacing (11) by an unqualified identification with the whole residue-field representation would overlook its torsion contribution.
5. A sharp lower bound
Choose a finite field of characteristic different from containing a primitive th root . For example, for any prime one may use . Set and
for this matrix has no columns. Define by
On the space of all -valued functions on define
Then .
Lemma 5.1.
If has inner rank at most , then .
Proof.
For any summand in (15) to be nonzero, we must have
If these conditions fail, every summand vanishes. If they hold, write . Substitution in (14) gives
This is zero unless . Together with (16), that equality forces to be invertible: all columns other than the second are the corresponding standard basis vectors, and the second coordinate of the second column is one. But , a contradiction.
Let be the clonoid generated by . Every member of its -ary part is a -linear combination of substitutions of by matrices with input columns. Hence every -ary function generated by is a -linear combination of functions with . This follows directly by combining precompositions into a factorization through variables and postcompositions into linear combinations. Lemma 5.1 shows that vanishes on all such functions. Since , the clonoid is not generated by its -ary part. This completes the proof of Theorem 1.1.
6. Checks and further questions
The proof above is independent of computation. The accompanying standard-library script check.py provides exact finite diagnostics of the Fourier projector definition, the nonzero-character sandwich, the zero-character quotient calculation, and the lower-bound functional. It uses arithmetic in finite prime fields, including targets containing the required roots of unity.
The lower-bound check includes all matrices in for , and all matrices in for . It also checks one lift of every residue matrix in and additional specified cases. These checks distinguish singular reduction from inner rank: the functional is tested on the larger set of matrices with singular reduction, which contains every matrix of the forbidden inner rank. They do not establish an infinite theorem by enumeration.
The method isolates why an elementary abelian factor costs one variable, while the cyclic factor can cost two: a nonzero congruence character is a rank-one matrix, and its nilpotent form requires two fixed coordinates. For sources with more than one cyclic factor, higher-rank congruence characters occur. The reduction in Lemma 3.1 then needs a different argument. Groups of higher exponent also require additional congruence layers. Neither case is settled here, and the general finite-module conjecture remains outside the scope of this paper.
References
[1] S. Fioravanti, M. Kompatscher and B. Rossi, Clonoids over vector spaces, preprint (2026), arXiv:2602.04034v1.
[3] L. G. Kovács, Semigroup algebras of the full matrix semigroup over a finite field, Proceedings of the American Mathematical Society 116 (1992), 911–919. doi:10.1090/S0002-9939-1992-1123658-6.
[4] N. J. Kuhn, Generic representation theory of finite fields in nondescribing characteristic, Advances in Mathematics 272 (2015), 598–610. doi:10.1016/j.aim.2014.12.012; arXiv:1405.0318v2.