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Polynomial Completeness and Conjugation
-by Matt Insall and Ralph W. Wilkerson
2000 Mathematics subject classification (Amer. Math. Soc.): 08 A 40, 03 C 05
O.
Introduction. We consider a “generalization” of the notion of polynomial completeness for algebraic structures,
and show the following results:
Theorem 1: Let A = (A; (fi)iI) be an algebra in which every A-valued function defined on any finite power of the set A
is the conjugate of some polynomial function of A. If A is finite, then the algebra A is polynomially complete.
Theorem 2: Let A = (A; (fi)iI) be an algebra, such that every unary function defined on the set A is the conjugate of
some polynomial function of A. If A is countable and the algebra A has only countably many fundamental operations,
then A is finite.
The proof of theorem 1 is due to Kearnes and Pixley [KP], but seems to have not appeared in print before now.
Theorem 2 implies a well-known result, that every algebra with countably many fundamental operations which is both
countable and polynomially complete is necessarily finite. Our proof of this well-known result seems new. Together,
theorems 1 and 2 yield that, at least for countable algebras with countably many operations, the “generalization” of
polynomial completeness, obtained by only requiring that all functions be conjugates of polynomial functions, reduces
to the notion of polynomial completeness for finite algebras.
1.
Proofs of Theorems 1 & 2. Let us now prove the theorems stated above. By a conjugate of a polynomial
function of an algebra A we mean a function f : An  A such that, for some n-ary member p of the polynomial clone
of A, and for some bijection g on the set A, we have f = g-1 ○ p ○ g(n), where g(n) denotes the bijection on An defined by
the formula
g(n)(a1,…,an) = (g(a1),…,g(an)).
Of course, if n = 1, this notion reduces to the usual notion of a conjugate: f = g-1 ○ p ○ g.
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Proof of Theorem 1[5]: Let s be a Sheffer function on the finite set A. It is easy to see that any conjugate of a
Sheffer function is again a Sheffer function, so if s is a conjugate of a polynomial function p of A, then p is itself a
Sheffer function, so that the polynomial clone of A includes all functions on A. That is, the algebra A is polynomially
complete. 
The proof of theorem 2 will, as stated before, use the fact that any countable algebra with only countably many
fundamental operations is 1-polynomially equivalent to a multi-unary algebra. In fact, the 1-clone, P1 = {f:AA | f is a
polynomial of A}, of such an algebra A may be assumed to be the collection of fundamental operations of A, and we
make this assumption in what follows.
Proof of Theorem 2: Assume, as suggested, that the algebra A is multi-unary, with every unary polynomial of A a
fundamental operation:
A = (A;f1, f2, …, fn, …),
where the set A is denumerable. Let F = AA be the set of all functions from A into A, let B = {fF | f is bijective}, and
let
D = {fB | (∀ xA), ( ∃nN ) (f(n) (x) = x and (∀ nN\xA | f(n) (x) =x, but (∀ j<n) (f(j) (x) ≠ x)} is finite}.
Define an equivalence relation ≈ on the sets F, B and D by conjugation:
f ≈g if and only if there is an h  B such that h-1
○f ○
h = g.
We claim that the factor set D/≈ = {[f]≈ | f  D} is uncountable. (Here, as usual, [f]≈ = {gD | g ≈ f}.) To see this, for
each f in D, let γf be the countable sequence of integers γf = (k1, k2, …, kn, …), where for each n  N\{0},
kn = |{xA | f(n) (x) = x, but (∀j<n) (f(j)(x) ≠ x)}|
Note that for f, g, in D, we have that f ≈ g if and only if γf = γg. Since the set
NN = {(k1, k2, …, kn, …) | (∀j  N \{0}) (kj  N)}
Is uncountable, and each γ NN corresponds to a unique member of D/≈, it follows that D/≈ is uncountable.
Now, the assumptions on the algebra A yield the containment
| j  N\{0}},
D ⊆ U{[fj]
because D ⊆ B ⊆ F, so that D/≈ is countable, a contradiction. Thus A is finite, as claimed. 
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Now, we state without proof a corollary of theorem 2, which demonstrates our claim in the introduction that for
countable algebras with countably many operations, the “generalization” of polynomial completeness, obtained by
only requiring that all functions be conjugates of polynomial functions, reduces to the notion of polynomial
completeness for finite algebras.
Corollary 3:
Let A be a primal or polynomially complete countable algebra with countably many fundamental
operations. Then A is finite.
2.
Directions of further enquiry. The result in theorem 1 has the property that the conditions imposed are on
polynomials of all arities. It remains to consider the implications of imposing only restricted forms of this variation on
the polynomial completeness theme. In [6], related results in certain finite rings were presented. The computational
complexity of the general problem in this case, of characterizing the unary functions on a ring that are conjugates of
polynomial functions is very high.
(A brute force algorithm to completely classify them tests or generates
approximately nn functions on an n-element ring.) In [6], we reported that there are 64 polynomial functions on the
ring of integers modulo 4, and there are 144 non-polynomial functions that are conjugates of polynomial functions.
As can be checked using the Universal Algebra calculator developed by E. Kiss and R. Freese [8], there are, on the
ring of integers modulo 6, exactly 108 unary polynomial functions. At the time of publication of [6], it was learned that
in the ring, Z4, of integers modulo 4, a unary function f is a polynomial function if and only if both f(2) is in {f(0), f(0)+2}
and f(3) is in {f(1), f(1) + 2}. This characterization is useful in testing unary functions for conjugacy to a polynomial
function. (In this case, one need only consider cases corresponding to 4!=24 conjugacies, to find and characterize or
classify the conjugates of polynomial functions.) Conditions of a similar nature for a unary function to be a polynomial
function on the ring, Z6, of integers modulo 6, are currently being developed, and will be used to help develop a
procedure for finding all conjugates of polynomial functions on this ring.
However, it seems that there is not likely to
be a reasonable pattern in the lists of properties that characterize those unary functions that are polynomial functions
on a given finite ring, even in the class of rings of integers modulo n, for some given integer. Computation using
programs like that of Kiss and Freese [8] can be helpful, and we will continue to extend the computational results in
the future.
In another direction, but again in the case of (finite) rings, it is not difficult to prove the following result:
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Theorem 3: Let R be a ring. Then either R is a finite field or there is a unary function f on R that is not a polynomial
function on R.
We suspect that also the following result holds, which, by its very nature, is more difficult to prove:
Conjecture: Let R be a ring. Then either R is a field or there is a unary function f that is not the conjugate of a
polynomial function on R.
At first glance, this may appear to be a consequence of theorem 1 above. However, in theorem 1, it was assumed
that the functions of all arities are conjugates of polynomial functions. This allows the use of sheffer functions to
complete the argument in the finite case, because inclusion of higher arities includes as conjugates of polynomial
functions some sheffer function, but restriction here to arity one keeps sheffer functions out of the picture, since there
are no unary sheffer functions. When some higher arity functions are not polynomial functions, it is concievable that
still all unary functions are conjugates of polynomial functions, while not being polynomial functions themselves. In
the infinite case, however, the situation may be more complicated, due to the fact that a natural argument in that case
is a cardinality argument, and depends upon a foundational consideration, namely the question of whether on a given
infinite set X, the set of bijections is of strictly larger cardinality than the cardinality of X. See [7], pages 83 and 84.
Acknowledgement. We would like to thank Professor Rick Smith of the University of Florida for valuable suggestions we were
able to use in our argument for Theorem 2.
Bibliography
[1] S. Burris and H.P. Sankapanavar, A Course in Universal Algebra. New York, N.Y.: Springer-Verlag, 1980.
[2]
P.M. Cohn, Universal Algebra. New York, N.Y.: Harper & Row, 1965.
[3]
G. Gratzer, Universal Algebra. New York: Springer-Verlag, 1979.
[4]
D. Hobby and R. McKenzie, The Structure of Finite Algebras. AMS Series in Contemporary Mathematics,
v.76. Providence, R.I.: American Mathematical Society, 1988.
[5] K. Kearnes and A.F. Pixley, Private Communication.
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[6] L. Mullin, M. Insall and R. Wilkerson, ``Conjugating Polynomials on Finite Rings'', in the 1994 ACM SigApp
Symposium on Applied Computing.
[7] P. Howard and J. Rubin, ``Consequences of the Axiom of Choice'', Mathematical Surveys and Monographs,
Amer. Math. Soc., vol. 59, 1998.
[8] E. Kiss and R. Freese, ``An Algebra Calculator Program'',
http://www.cs.elte.hu/~ewkiss/software/uaprog/index.html
Matt Insall
Department of Mathematics and Statistics
University of Missouri at Rolla
Rolla, Missouri 65409-0020
USA
Ralph W. Wilkerson
Department of Computer Science
University of Missouri at Rolla
Rolla, Missouri 65409
USA
Typed by Lois Jaquess using Microsoft Word
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