Polynomial Identities for Bernstein Algebras of Simple Mendelian ...

Polynomial Identities for Bernstein Algebras of
Simple Mendelian Inheritance
Murray R. Bremner ⋆,⋆⋆ , Yunfeng Piao, and Sheldon W. Richards
Department of Mathematics and Statistics
University of Saskatchewan, Saskatoon, Canada
Abstract. The Bernstein algebras D n Gα of simple Mendelian inheritance
are obtained by n-fold duplication of the gametic algebra Gα
whose basis consists of the α alleles A1, . . . , Aα with structure constants
AiAj = 1 1
Ai + Aj. We demonstrate the fundamental role played by the
2 2
recombination identity R of degree 4 in the theory of Bernstein algebras.
We simplify and generalize results of Bernad et al. by showing that every
polynomial identity for Gα is a consequence of R and another identity Q
of degree 4, and that every polynomial identity for the zygotic algebra
DGα is a consequence of R. We use computer algebra to determine the
polynomial identities of degree ≤ 7 for the copular algebra D 2 Gα; in this
case the values of R produce absolute zero-divisors.
1 Introduction
Hardy [11] and Weinberg [26] independently found the equations describing the
distribution of genotypes for Mendelian inheritance in a randomly mating population,
and observed that stability is achieved in the second generation. The
problem of finding all coefficients of heredity for which the probability distribution
of the second generation remains unchanged in subsequent generations was
posed and partially solved by Bernstein [4]. At the same time but independently,
the nonassociative structures called genetic algebras were introduced by Etherington
[8, 9]. The Bernstein problem was reformulated using abstract algebra
by Lyubich [18], and the formal definition of Bernstein algebras was given by
Holgate [14]; the generalization to n-th order Bernstein algebras was made by
Abraham [1]. Unlike most nonassociative structures, Bernstein algebras are defined
by properties of the weight function rather than by polynomial identities.
Surveys of the theory of genetic algebras can be found in Wörz-Busekros [27],
Lyubich [19] and Reed [23].
Holgate [13] demonstrated that the gametic and zygotic algebras of simple
Mendelian inheritance satisfy the Jordan identity. Walcher [25] studied the
conditions under which a (first-order) Bernstein algebra is power-associative or
Jordan. Mártinez [20] used free Jordan Bernstein algebras to obtain results on
the degree of nilpotency of the ideal of weight zero elements. Bernad et al. [2, 3]
⋆ Corresponding author. Email: bremner@math.usask.ca
⋆⋆ Supported by NSERC (Natural Sciences and Engineering Research Council)

2 Bremner, Piao and Richards
found a complete set of identities for the gametic and zygotic algebras of simple
Mendelian inheritance on two alleles. López-Díaz et al. [17] studied Bernstein
algebras which satisfy the Glennie identities for special Jordan algebras. Correa
et al. [7] (see also Peresi [22]) used computer algebra to determine the minimal
polynomial identities satisfied by all Bernstein algebras, as well as normal and
exceptional algebras. Hentzel et al. [12] used computer algebra to calculate the
index of solvability of the ideal of all weight zero elements. Polynomial identities
for second-order Bernstein algebras have been studied by Gónzalez et al.
[10] and Labra et al. [15]. The operation of intermolecular recombination was
formalized by Landweber and Kari [16], and its polynomial identity R of degree
4 was discovered by Bremner [6]. It has recently been shown by Sverchkov [24]
that any algebra satisfying R is a special Jordan algebra, and that R implies all
the identities satisfied by intermolecular recombination.
In Sect. 2 we recall basic results on nonassociative algebras and their polynomial
identities; we introduce the gametic, zygotic and copular algebras of simple
Mendelian inheritance on a finite number of alleles. In Sect. 3 we consider the
gametic algebras; we simplify and generalize Theorem 2.1 of [3] by reducing the
number of identities to two (the recombination identity R and another identity
Q of degree 4) and proving that they hold for any number of alleles. In Sect. 4
we consider the zygotic algebras; we simplify and generalize Theorem 3.2 of [3]
by reducing the number of identities to one (the identity R) and proving that it
holds for any number of alleles. In Sect. 5 we use computer algebra to determine
the identities of degree ≤ 7 satisfied by the copular algebras: the first identities
which do not follow from commutativity occur in degree 5; one of them states
that the values of R give absolute zero-divisors; there are no new identities in
degrees 6 or 7; unlike the gametic and zygotic algebras, the identities for two
alleles differ from the identities for three or more alleles.
2 Preliminaries
Definition 1. An algebra A is a vector space over a field F with a bilinear
product m: A × A → A, denoted (x, y) ↦→ m(x, y) = xy. We say that A is
commutative if xy = yx for all x, y ∈ A.
Definition 2. A weight function on an algebra A is a nonzero linear map
ω : A → F which satisfies ω(xy) = ω(x)ω(y) for all x, y ∈ A. An algebra with a
weight function is called a baric algebra.
Definition 3. If A is commutative then the duplicate DA is the algebra with
underlying vector space S 2 (A) and product (x1 ⊗ x2)(y1 ⊗ y2) = x1x2 ⊗ y1y2
extended bilinearly; note that we do not define (x1 ⊗ x2)(y1 ⊗ y2) = x1y1 ⊗ x2y2.
Lemma 4. If A is commutative then DA is commutative. If A is a baric algebra
with weight function ω, then DA is a baric algebra whose weight function is the
bilinear extension of ω(x1 ⊗ x2) = ω(x1)ω(x2).
Proof. Wörz-Busekros [27, Theorem 6.15, page 116]. ⊓⊔

Polynomial Identities for Bernstein Algebras 3
Definition 5. The plenary powers of an element x ∈ A are defined by x [0] = x
and x [n+1] = x [n] 2 for n ≥ 0. (Some authors write x [n+1] for our x [n] .)
Definition 6. A Bernstein algebra of order n is a commutative baric algebra
A which satisfies x [n+1] = ω(x) 2nx
[n] for all x ∈ A.
Lemma 7. If A is a Bernstein algebra of order n then its duplicate DA is a
Bernstein algebra of order n+1.
Proof. Ouattara [21, Theorem 2.1]. ⊓⊔
Definition 8. The gametic algebra Gα of simple Mendelian inheritance on α
alleles is the commutative baric algebra with natural basis A1, A2, . . . , Aα, product
AiAj = 1
2Ai + 1
2Aj
and weight function ω i xiAi
= i xi. The canonical
basis is C1 = A1, Ci = Ai − A1 (i ≥ 2).
Lemma 9. The canonical basis of Gα satisfies C2 1 = C1, C1Ci = 1
2Ci (i ≥
2), CiCj = 0 (i, j ≥ 2) and ω i xiCi
= x1. The gametic algebra Gα is a
Bernstein algebra of order 0.
Proof. One easily verifies that i aiCi
2
= a1 i aiCi. ⊓⊔
Definition 10. The duplicate of the gametic algebra Gα is the zygotic algebra
DGα; the duplicate of the zygotic algebra is the copular algebra D 2 Gα.
Duplication can be repeated indefinitely; Lemmas 7 and 9 imply that D n Gα is a
Bernstein algebra of order n.
Lemma 11. We have dim D n+1 Gα = dim D n Gα+1
2
. In particular,
dim Gα = α , dim DGα = 1
2 α(α+1) , dim D2 Gα = 1
8 α(α+1)(α2 +α+2) .
Proof. Straightforward. ⊓⊔
Definition 12. A nonassociative monomial is a fully parenthesized string of
variables; a nonassociative polynomial is a linear combination of nonassociative
monomials. A nonassociative polynomial is homogeneous if every monomial
has the same degree; it is multilinear if each variable occurs exactly once
in every monomial. A nonassociative polynomial I(x1, . . . , xn) is a polynomial
identity for the algebra A if I(x1, . . . , xn) = 0 for all x1, . . . , xn ∈ A.
Lemma 13. Any polynomial identity over a field of characteristic 0 is equivalent
to a finite set of homogeneous multilinear identities.
Proof. Zhevlakov et al. [28, Chapter 1]. ⊓⊔
Remark 14. Since D n Gα is commutative, we consider only homogeneous multilinear
commutative nonassociative monomials. In a given degree d, such a monomial
consists of a commutative association type (arrangement of parentheses)
and a permutation of the d variables a1, . . . , ad. (See Example 16 below.)

4 Bremner, Piao and Richards
Definition 15. We impose a total order on association types. For i =
1, 2 let ti be commutative association types in degree d, let xi be the monomial
consisting of the string a1 · · · ad with association type ti, and let xi = yizi be
factorizations where yi and zi have association types t ′ i and t′′ i respectively. These
factorizations are unique if we impose the conditions (i) deg yi > deg zi and
(ii) if deg yi = deg zi then t ′ i precedes t′′ i in the total order already defined on
association types of lower degree. We then say that t1 precedes t2 if and only if
(i) deg x1 > deg x2 or (ii) deg x1 = deg x2 and t ′ 1 precedes t ′ 2.
Example 16. The ordered commutative association types in degrees 3 to 6 are
(ab)c , ((ab)c)d , (ab)(cd) ,
(((ab)c)d)e , ((ab)(cd))e , ((ab)c)(de) .
(((ab)c)d)e)f , (((ab)(cd)e)f , ((ab)c)(de))f ,
((ab)c)d)(ef) , ((ab)(cd)(ef) , ((ab)c)((de)f) .
Lemma 17. The number Cd of commutative association types in degree d is
given by this formula, where the binomial coefficient occurs only for even d:
C1 = 1 , Cd =
⌊(d−1)/2⌋
Cd/2 + 1
Cd−iCi +
.
2
i=1
Proof. Straightforward. ⊓⊔
Definition 18. We impose a total order on the monomials in each association
type. Any type t in degree d has a number s of symmetries arising from
commutativity, giving d!/2 s equivalence classes of monomials with type t. As representative
of each class we choose the monomial whose underlying permutation
comes first in lexicographical order.
Example 19. In degree 3 there is one symmetry, (ab)c = (ba)c, giving three inequivalent
monomials: (ab)c, (ac)b, (bc)a. In degree 4, the first type has one
symmetry, ((ab)c)d = ((ba)c)d, and the second type has three symmetries,
(ab)(cd) = (ba)(cd) = (ab)(dc) = (cd)(ab), giving 12 + 3 = 15 monomials:
((ab)c)d , ((ab)d)c , ((ac)b)d , ((ac)d)b , ((ad)b)c , ((ad)c)b ,
((bc)a)d , ((bc)d)a , ((bd)a)c , ((bd)c)a , ((cd)a)b , ((cd)b)a ,
(ab)(cd) , (ac)(bd) , (ad)(bc) .
Lemma 20. In degree d, the number of homogeneous multilinear commutative
nonassociative monomials is (2d−3)!! where k!! = (k)(k−2) · · · (3)(1) for odd k.
Proof. Bremner [5, Proposition 1]. (The same proof holds in the commutative
case since the monomials are multilinear.) ⊓⊔

Polynomial Identities for Bernstein Algebras 5
Definition 21. We write Pd for the space of nonassociative polynomials:
the vector space with basis consisting of the (homogeneous multilinear commutative
nonassociative) monomials of degree d. The symmetric group Sd acts on the
monomials: omitting the association type, we set σ(ai1 · · · aid ) = a σ(i1) · · · a σ(id);
we then replace the monomial by the standard representative of its equivalence
class. This action of Sd does not change the association types and gives Pd the
structure of an Sd-module.
Lemma 22. The subspace I(A) ⊆ Pd consisting of the polynomial identities
satisfied by the algebra A is an Sd-submodule of Pd.
Proof. If I is an identity for A then so is σI, and any linear combination of
identities for A is again an identity for A. ⊓⊔
Definition 23. Let I(a1, . . . , ad) be an element of Pd. We introduce a new indeterminate
ad+1 and define d + 1 elements of Pd+1, the liftings of I to degree
d + 1: applying multiplication, and substitution for i = 1, . . . , d, we obtain
I(a1, . . . , ad)ad+1 , and I(a1, . . . , aiad+1, . . . , ad) for 1 ≤ i ≤ d .
The liftings of I generate the Sd+1-submodule of Pd+1 consisting of the identities
in degree d + 1 which follow from I.
3 Gametic Algebras
From now on, except as indicated, F is the field Q of rational numbers.
Definition 24. We define polynomials I, J ∈ P4 and K ∈ P5:
I = 2((ab)c)d − 2((ac)b)d − ((ad)b)c + ((ad)c)b ,
J = ((ab)c)d + ((ab)d)c − ((ac)b)d − ((ad)b)c + ((cd)a)b − (ab)(cd) ,
K = 2(((ab)c)d)e − 2(((ac)b)d)e − ((ab)c)(de) + ((ac)b)(de) .
Theorem 25. Every identity satisfied by G2, the gametic algebra on two alleles,
is a consequence of commutativity, I ≡ 0, J ≡ 0 and K ≡ 0.
Proof. Bernad et al. [3, Theorem 2.1]. ⊓⊔
Definition 26. We define polynomials Q, R ∈ P4:
Q = 3((ab)c)d − ((ab)d)c − 3((ac)b)d + ((ac)d)b ,
R = 2((ab)c)d − ((ab)d)c − ((ac)b)d − ((bc)a)d + (ab)(cd) .
R is the recombination identity satisfied by the operation of intermolecular
recombination from the theory of formal languages and DNA computing; see
Landweber and Kari [16], Bremner [6], Sverchkov [24].
Lemma 27. I and J generate the same S4-submodule of P4 as Q and R.

6 Bremner, Piao and Richards
Proof. It suffices to show that I and J can be expressed as linear combinations
of permutations of Q and R, and conversely:
4I = 3Q(a, b, c, d) + Q(a, b, d, c) − Q(a, c, d, b) ,
8J = 6Q(a, b, c, d) + 3Q(a, b, d, c) + Q(a, c, d, b) + 3Q(b, a, c, d) − 3Q(c, a, d, b)
− Q(c, b, d, a) − 8R(a, b, c, d) ,
Q = 2I(a, b, c, d) − I(a, b, d, c) + I(a, c, d, b) ,
2R = 2I(a, b, c, d) + I(a, c, d, b) + 2I(b, a, c, d) − I(b, a, d, c) + I(b, c, d, a)
− I(c, a, d, b) − 2J(a, b, c, d) .
These equations can be easily verified. (Each monomial must be “straightened”
by applying commutativity: replacing it by its standard representative.) ⊓⊔
Lemma 28. The identity K is a consequence of the identity Q.
Proof. We can express K as a linear combination of the liftings of Q:
48K = 36 Q(a, b, c, d)e + 12 Q(a, b, d, c)e + 21 Q(a, b, e, c)d − 15 Q(a, b, e, d)c
− 12 Q(a, c, d, b)e − 21 Q(a, c, e, b)d + 15 Q(a, c, e, d)b + 7 Q(a, d, e, b)c
− 7 Q(a, d, e, c)b − 21 Q(b, a, e, c)d + 15 Q(b, a, e, d)c + 17 Q(b, c, e, a)d
− 8 Q(b, c, e, d)a − 3 Q(b, d, e, a)c + 21 Q(c, a, e, b)d − 15 Q(c, a, e, d)b
− 17 Q(c, b, e, a)d + 8 Q(c, b, e, d)a + 3 Q(c, d, e, a)b + 9 Q(d, a, e, b)c
− 9 Q(d, a, e, c)b + 3 Q(d, b, e, a)c − 3 Q(d, c, e, a)b + 25 Q(ae, b, c, d)
+ 20 Q(ad, b, c, e) − 9 Q(ae, b, d, c) + 12 Q(ad, b, e, c) + 9 Q(ae, c, d, b)
− 12 Q(ad, c, e, b) − 24 Q(be, a, c, d) + 8 Q(be, a, d, c) + 24 Q(ce, a, b, d)
− 8 Q(ce, a, d, b) + 6 Q(a, de, b, c) − 6 Q(a, de, c, b) − 18 Q(a, b, c, de) .
This equation can be verified by lengthy hand calculations, or with a computer
algebra system. There are 36 terms on the right side of the equation; each expands
into a linear combination of 4 monomials. Of the resulting 144 monomials,
24 must be replaced by the standard representative of the equivalence class; then
only 52 distinct monomials remain in the expansion. Let C be the matrix of size
52 × 36 in which Cij is the coefficient of monomial i in the expansion of term j.
Summing over the rows of C we obtain the left side of the equation. ⊓⊔
Remark 29. The equation of Lemma 28 was found as follows. There are 5 liftings
of Q to degree 5: Q(a, b, c, d)e, Q(ae, b, c, d), Q(a, be, c, d), Q(a, b, ce, d) and
Q(a, b, c, de); and five liftings of R. We construct a matrix L of size (105+120) ×
105, initialized to zero. For each lifting, we apply the 120 permutations of the
variables, store the results in the rows of the lower 120 × 120 block of L, and
compute the row canonical form; the lower block is now zero. The final rank is
100: this is the dimension of the S5-submodule U ⊂ P5 consisting of identities in
degree 5 implied by Q and R. Only five liftings increase the rank: U is generated
by Q(a, b, c, d)e, Q(ae, b, c, d), Q(a, be, c, d), Q(a, b, c, de) and R(a, b, c, d)e. (See

Polynomial Identities for Bernstein Algebras 7
Algorithm 40 below for a formalization of this procedure.) We construct a matrix
of size 105 × 600 consisting of blocks of size 105 × 120. In column j of block k
we put permutation j of generator k. We compute the row canonical form; each
column containing the leading 1 of a row corresponds to a basis vector of the
column space. This gives 100 permutations of the generators forming a linear
basis of U. We form a matrix of size 105 × 101 which has these basis elements
in columns 1–100 and identity K in column 101. We compute the row canonical
form and obtain the coefficients of K as a linear combination of permutations
of liftings of Q and R. (Note that R does not appear in the final result.)
Theorem 30. The gametic algebras satisfy the same identities: for all α, every
identity for Gα is a consequence of commutativity, Q ≡ 0 and R ≡ 0.
Proof. It follows from Theorem 25 and Lemmas 27 and 28 that every identity
for Gα in the case α = 2 is a consequence of Q and R. Since G2 ⊆ Gα for all α,
every identity for Gα is an identity for G2, and so in each degree the identities
for Gα are a subspace of the identities for G2. It remains to show the converse:
that Gα satisfies Q and R. It suffices to consider basis elements since Q and R
are multilinear. Evaluating the association types in degree 4 we obtain
Aℓ = 1
AiAj Ak
8Ai + 1
8Aj + 1
4Ak + 1
2Aℓ ,
1
AiAj AkAℓ = 4Ai + 1
4Aj + 1
4Ak + 1
4Aℓ .
From this we obtain
Q
1 1 1
Ai, Aj, Ak, Aℓ = 3 8Ai+ 8Aj+ 4
− 3 1 1 1 1
8Ai+ 4Aj+ 8Ak+ 2Aℓ
1 + 8
R
1 1 1
Ai, Aj, Ak, Aℓ = 2 8Ai+ 8Aj+ 4
− 1 1 1 1
8Ai+ 4Aj+ 8Ak+ 2Aℓ
1 − 4
+ 1 1 1 1
4Ai+ 4Aj+ 4Ak+ 4Aℓ
.
1 Ak+ 2Aℓ 1 Ai+ 2
1 Ak+ 2Aℓ Ai+ 1
8
1 1 1
− 8Ai+ 8Aj+ 2
1 1
Aj+ 8Ak+ 4Aℓ
,
1 1 − 8Ai+ 8
Aj+ 1
8
Ak+ 1
2 Aℓ
1 Ak+ 4Aℓ
1 1
Aj+ 2Ak+ 4Aℓ
In both results all terms cancel. ⊓⊔
4 Zygotic Algebras
The natural basis of DGα is Aij = Ai ⊗ Aj (1 ≤ i ≤ j ≤ g). Definition 3 gives
AijAkℓ = 1
4 Aik + 1
4 Aiℓ + 1
4 Ajk + 1
4 Ajℓ .
Definition 31. We define polynomials L, M, N (degree 4) and O, P (degree 5);
note that L is the Jordan identity and that L, M, O are not multilinear:
L = ((a 2 )b)a − a 2 (ab) , M = −2((ab)b)a + (b 2 a)a + a 2 b 2 ,
N = ((ab)c)d − ((ab)d)c − ((bc)a)d + ((bc)d)a + ((bd)a)c − ((bd)c)a ,
O = −2(((ab)c)c)d + 2(((bc)a)c)d − ((c 2 b)a)d + ((ab)c 2 )d ,
P = (((ab)c)d)e − (((ad)c)b)e − (((bc)a)d)e + (((cd)a)b)e .

8 Bremner, Piao and Richards
Theorem 32. Every identity satisfied by DG2, the zygotic algebra on two alleles,
follows from commutativity, L ≡ 0, M ≡ 0, N ≡ 0, O ≡ 0 and P ≡ 0.
Proof. Bernad et al. [3, Theorem 3.2]. ⊓⊔
Lemma 33. L, M and N are consequences of R, and conversely.
Proof. We easily verify that
L = −R(a, a, a, b) , M = R(a, a, b, b) + R(a, b, a, b) − R(a, b, b, a) ,
4N = 4R(a, b, c, d) + 4R(a, b, d, c) − 4R(a, c, d, b) + 3R(d, a, b, c) − 3R(d, a, c, b)
+ 5R(d, b, a, c) − R(d, b, c, a) − 5R(d, c, a, b) − 3R(d, c, b, a) .
We now linearize L and M, and make the leading coefficients 1, obtaining
L(a, b, c, d) = ((ac)b)d + ((ad)b)c + ((cd)b)a − (ab)(cd) − (ac)(bd) − (ad)(bc) ,
M(a, b, c, d) = ((ab)d)c + ((ad)b)c + ((bc)d)a − ((bd)a)c − ((bd)c)a + ((cd)b)a
− 2(ac)(bd) ,
2R(a, b, c, d) = −2L(a, b, c, d) + L(a, c, b, d) − L(a, d, b, c) + M(a, b, c, d)
+ M(a, b, d, c) + 2N(a, b, c, d) − N(a, c, b, d) + N(a, d, b, c) .
The last equation can be easily verified. ⊓⊔
Lemma 34. O and P are consequences of R.
Proof. We have the equations
O = 3R(a, b, d, c)c−R(a, c, b, c)d−4R(a, c, d, b)c+R(a, c, d, c)b+2R(a, d, b, c)c
−2R(a, d, c, b)c+R(b, c, a, c)d+R(b, c, c, a)d+3R(b, d, a, c)c+R(b, d, c, a)c
−2R(c, d, a, b)c−R(c, d, a, c)b−R(c, d, b, a)c−R(ab, c, c, d)+R(ab, c, d, c)
+R(ac, b, c, d)−R(ac, b, d, c)+R(ac, d, b, c)−R(ac, d, c, b)−R(cd, a, b, c)
+R(cd, a, c, b) ,
4P = 4R(a, b, c, d)e+4R(a, b, d, c)e−4R(a, c, d, b)e+3R(a, d, b, c)e−5R(a, d, c, b)e
+6R(a, d, e, b)c+2R(a, e, b, d)c+6R(a, e, d, b)c+R(b, d, a, c)e+3R(b, d, c, a)e
−8R(b, d, e, a)c+8R(b, d, e, c)a−2R(b, e, a, d)c+2R(b, e, c, d)a−4R(b, e, d, a)c
+4R(b, e, d, c)a−3R(c, d, a, b)e−3R(c, d, b, a)e−6R(c, d, e, b)a−2R(c, e, b, d)a
−6R(c, e, d, b)a+4R(d, e, a, b)c−4R(d, e, b, a)c+4R(d, e, b, c)a−4R(d, e, c, b)a
−2R(ad, b, c, e)+2R(ad, b, e, c)+2R(bd, a, c, e)−2R(bd, a, e, c)−2R(bd, c, a, e)
+2R(bd, c, e, a)+2R(bd, e, a, c)−2R(bd, e, c, a)−2R(be, d, a, c)+2R(be, d, c, a)
−2R(cd, b, e, a)+2R(cd, b, a, e) .
These can be verified by tedious but straightforward calculation. ⊓⊔
Theorem 35. The zygotic algebras satisfy the same identities: for all α, every
identity for DGα is a consequence of commutativity and R.

Polynomial Identities for Bernstein Algebras 9
Proof. It follows from Theorem 32 and Lemmas 33 and 34 that every identity for
DGα in the case α = 2 is a consequence of R. Since DG2 ⊆ DGα for all α, every
identity for DGα is an identity for DG2, and so in each degree the identities for
DGα are a subspace of the identities for DG2. It remains to show the converse:
that DGα satisfies R. In the natural basis, we ignore the assumption that i ≤ j
and identify Aij with Aji. We first compute the general product in degree 3:
AijAkℓ Aim+Ain+Ajm+Ajn+Akm+Akn+Aℓm+Aℓn .
Amn = 1
8
For the first association type in degree 4, we have
AijAkℓ Amn Apq =
1
1
1
16
Aip+Aiq+Ajp+Ajq + 16
Akp+Akq+Aℓp+Aℓq + 8
Amp+Amq+Anp+Anq ,
and therefore
2
AijAkℓ Amn Apq −
AijAkℓ Apq Amn −
AijAmn Akℓ Apq
−
AkℓAmn Aij Apq = − 1
16
Aim + Ain + Ajm + Ajn + Aip + Aiq
.
+ Ajp + Ajq + Akm + Akn + Aℓm + Aℓn + Akp + Akq + Aℓp + Aℓq
For the second association type in degree 4, we have
1
AijAkℓ AmnApq = 16
Aim + Ain + Aip + Aiq + Ajm + Ajn
+ Ajp + Ajq + Akm + Akn + Akp + Akq + Aℓm + Aℓn + Aℓp + Aℓq
Adding the last two results gives R
Aij, Akℓ, Amn, Apq ≡ 0. ⊓⊔
5 Copular Algebras
The natural basis of D2Gα consists of Aijkℓ = Aij ⊗ Akℓ with 1 ≤ i ≤ j ≤ α,
1 ≤ k ≤ ℓ ≤ α, and either i < k or i = k, j < ℓ. The structure constants are
AijkℓAmnop = 1
16
Aikmo+Aikmp+Aikno+Aiknp+Aiℓmo+Aiℓmp+Aiℓno
+Aiℓnp+Ajkmo+Ajkmp+Ajkno+Ajknp+Ajℓmo+Ajℓmp+Ajℓno+Ajℓnp
where we identify Aijkℓ, Aijℓk, Ajikℓ, Ajiℓk, Akℓij, Aℓkij, Akℓji, Aℓkji. Little is
known about identities for D 2 Gα even when α = 2; in particular, there is no
known generating set for the identities of D 2 G2 as for G2 and DG2. In this
section our proofs are primarily computational. We often use arithmetic modulo
p = 101 so that each matrix entry fits in one byte. Modular arithmetic is essential
in higher degrees when the matrices are large; it saves memory, but requires
reconstruction of rational coefficients from modular coefficients and independent
verification of the rational results.
Lemma 36. Every identity of degree ≤ 4 satisfied by D 2 Gα (α ≥ 2) is a consequence
of commutativity. In particular, D 2 Gα does not satisfy the identity R.
.
,

10 Bremner, Piao and Richards
Proof. Since any identity in degree d implies identities in degree d+1, it suffices
to prove that D 2 Gα has no identities in degree 4. Since D 2 G2 ⊆ D 2 Gα, the
identities for D 2 Gα are a subspace of the identities for D 2 G2, and it suffices to
show that D 2 G2 has no identities in degree 4. We can use modular arithmetic,
since non-existence of identities over IFp implies non-existence over Q. (If I is a
nontrivial identity over Q, we multiply by the lcm of the denominators of the
coefficients, then divide by the gcd of the coefficients to get a primitive identity,
and finally reduce mod p to get a nontrivial identity over IFp.) We abbreviate
the natural basis of D 2 G2 by U = A1111, V = A1112, W = A1122, X = A1212,
Y = A1222, Z = A2222. Using p = 101 we then have U 2 = U, UV = V U = 51V ,
UX = XU = 76W , V 2 = 76X, V X = XV = 38Y , X 2 = 19Z, and all other
products are 0. There are 15 ordered monomials Mi in degree 4 (Example 19),
so we need to find 15 linearly independent constraints on their coefficients xi.
Evaluating each monomial Mi with abcd = UUV V we obtain a multiple of W or
X; for example, ((ab)c)d = ((UU)V )V = 38X, (ab)(cd) = (UU)(V V ) = 19W .
We obtain these constraints:
60x4 + 60x6 + 60x8 + 60x10 + 19x13 = 0 ,
38x1 + 38x2 + 19x3 + 19x5 + 19x7 + 19x9 + 19x14 + 19x15 = 0 .
Setting abcd equal to UV UV , UV V U, V UUV , V UV U, V V UU, we get another
10 constraints, for a total of 12. Setting abcd equal to UUV X, UV UX, UV XU,
we find that every monomial produces a multiple of Y , giving another 3 constraints.
The resulting 15 × 15 coefficient matrix has full rank. ⊓⊔
Definition 37. The expansion matrix in degree d for a commutative nonassociative
algebra A of dimension r is a matrix Ed with (2d−3)!! + r rows and
(2d−3)!! columns, regarded as an upper block of size (2d−3)!! × (2d−3)!! and
a lower block of size r × (2d−3)!!. Column j corresponds to the j-th ordered
monomial of degree d.
Algorithm 38 (Fill and Reduce).
Input: Positive integers d (degree of identities), p (upper limit of pseudorandom
integers) and s (number of iterations required with stable rank). This
algorithm calls a procedure which evaluates the product in the algebra A of
dimension r whose elements are written as column vectors in a given basis.
Output: Once the rank has stabilized, the nullspace of the expansion matrix
contains the polynomial identities of degree d satisfied by the algebra A.
1. Initialize the expansion matrix Ed to zero.
2. Repeat until the rank of Ed has not increased for s iterations:
(a) Generate d pseudorandom elements x1, . . . , xd ∈ A: column vectors of
length r with components in {0, 1, . . . , p−1}.
(b) For j = 1, 2, . . . , (2d − 3)!!, evaluate the j-th monomial (a permutation
of a1, . . . , ad with an association type) for ak = xk (k = 1, . . . , d) and
store the result in column j of the lower block.
(c) Compute the row canonical form of Ed; the lower block is now zero.

Polynomial Identities for Bernstein Algebras 11
Definition 39. The generator matrix for commutative nonassociative identities
in degree d is a matrix Md with (2d−3)!! + d! rows and (2d−3)!! columns,
regarded as an upper block of size (2d−3)!! × (2d−3)!! and a lower block of size
d! × (2d−3)!!. Column j corresponds to the j-th ordered monomial of degree d.
Row i of the lower block corresponds to the i-th permutation of the d variables.
Algorithm 40 (Module Generators).
Input: An ordered set of Sd-module generators I1, . . . , Ik for a submodule
U ⊆ Pd of identities in degree d satisfied by the algebra A.
Output: The minimal ordered subset which generates U in the sense that each
element is not a member of the submodule generated by the previous elements.
1. Initialize the generator matrix Md to zero. Set oldrank = 0.
2. For ℓ = 1, . . . , k do:
(a) Set i = 0.
(b) For each σ ∈ Sd do:
i. Increment i.
ii. Apply σ to each monomial in Iℓ and replace the resulting monomial
by the standard representative of its equivalence class.
iii. Store the resulting permuted identity σIℓ in row i of the lower block.
(c) Compute the row canonical form of Md; the lower block is now zero.
(d) Set newrank = rank(Md).
(e) If newrank > oldrank then record Iℓ as a new generator.
(f) Set oldrank = newrank.
Definition 41. We define polynomials S, T ∈ P5:
S = ((ab)d)(ce)−((ab)e)(cd)+((ac)d)(be)−((ac)e)(bd)+((bc)d)(ae)−((bc)e)(ad),
T = ((ab)d)(ce)−((ac)e)(bd)+((ad)b)(ce)−((ae)c)(bd)−((bd)a)(ce)+((ce)a)(bd).
Theorem 42. Every identity of degree ≤ 7 satisfied by the copular algebras
D 2 Gα for α ≥ 3 is a consequence of commutativity, R(a, b, c, d)e = 0, S ≡ 0 and
T ≡ 0. (These identities are also satisfied by the copular algebra for α = 2.)
Proof. Lemma 36 shows that D 2 G3 satisfies no identities in degree ≤ 4. For the
computations in this proof we used the Maple package LinearAlgebra[Modular]
with p = 101. The nonzero products in D 2 G3 (mod p) are:
A1111A1111 = A1111 , A1111A1112 = 51A1112 , A1111A1113 = 51A1113 ,
A1111A1212 = 76A1122 , A1111A1213 = 76A1123 , A1111A1313 = 76A1133 ,
A1112A1112 = 76A1212 , A1112A1113 = 76A1213 , A1112A1212 = 38A1222 ,
A1112A1213 = 38A1223 , A1112A1313 = 38A1233 , A1113A1113 = 76A1313 ,
A1113A1212 = 38A1322 , A1113A1213 = 38A1323 , A1113A1313 = 38A1333 ,
A1212A1212 = 19A2222 , A1212A1213 = 19A2223 , A1212A1313 = 19A2233 ,
A1213A1213 = 19A2323 , A1213A1313 = 19A2333 , A1313A1313 = 19A3333 .

12 Bremner, Piao and Richards
Step 1: We show that D 2 G3 satisfies nontrivial identities in degree 5 over
IFp; we find a linear basis for the space of identities and a set of generators for
the S5-module of identities. There are 105 monomials in degree 5, and D 2 G3 has
dimension 21, so the expansion matrix has size 126 × 105. The first iteration of
Algorithm 38 generates these 5 pseudorandom elements of D 2 G3:
58 91 36 15 16 99 0 77 62 33 57 93 50 44 49 49 33 76 96 5 79
15 43 54 6 11 7 32 52 36 97 72 89 0 16 77 65 50 94 100 65 28
54 6 37 71 68 65 12 61 72 61 98 81 90 65 85 77 39 93 9 2 65
14 54 45 87 41 55 7 63 43 98 25 17 29 65 3 63 57 72 90 89 11
31 34 83 19 100 54 2 27 6 75 49 74 31 7 8 26 84 77 69 26 62
We evaluate the monomials on these elements, store the results in the lower block
of the expansion matrix, and compute the row canonical form; the rank is 21.
The second iteration generates another 5 elements, and again fills and reduces
the expansion matrix; the rank is 40. After the third iteration the rank is 45; we
perform another 100 iterations but the rank does not increase. It follows that
the space of identities has dimension 105 − 40 = 60. From the row canonical
form we extract the canonical basis for the nullspace. We sort these 60 identities
by increasing number of nonzero components. Using Algorithm 40 we find that
identity 1 generates an S5-submodule of dimension 45, identities 2–35 belong
to the submodule generated by identity 1, identity 36 increases the dimension
to 55, and identity 37 increases the dimension to 60 (the entire nullspace). It
follows that every identity of degree 5 follows from commutativity and identities
1, 36 and 37: these are R(a, b, c, d)e ≡ 0, S ≡ 0 and T ≡ 0.
Step 2a. From each identity I(a, b, c, d, e) in degree 5 we get six liftings:
I(a, b, c, d, e)f , I(af, b, c, d, e) , I(a, bf, c, d, e) ,
I(a, b, cf, d, e) , I(a, b, c, df, e) , I(a, b, c, d, ef) .
Applying Algorithm 40 to these 18 identities we find seven identities that generate
the same S6-module:
(R(a, b, c, d)e)f , R(af, b, c, d)e , R(a, b, c, d)(ef) , S(af, b, c, d, e) ,
S(a, b, c, df, e) , T (af, b, c, d, e) , T (a, bf, c, d, e) .
These identities cumulatively generate submodules of dimensions 270, 570, 705,
786, 826, 835 and 840.
Step 2b: There are 945 monomials in degree 6, so the expansion matrix has
size 966 × 945. The first 7 iterations of Algorithm 38 produce ranks 21, 41, 60,
75, 90, 102 and 105; we then perform another 100 iterations but the rank does
not increase. It follows that the S6-module of identities for D 2 G3 has dimension
945 − 105 = 840. Since this is also the dimension of the lifted identities, it
follows that every identity in degree 6 is a consequence of commutativity and
the identities in degree 5.
Step 3: We extend these computations to degree 7; the expansion matrix has
size 10416 × 10395. The 49 liftings of the 7 identities from degree 6 generate an

Polynomial Identities for Bernstein Algebras 13
S7-module of dimension 10185. The stable rank of the expansion matrix is 210,
and so the nullspace has dimension 10395 − 210 = 10185. Thus every identity in
degree 7 is a consequence of commutativity and the identities in degree 5.
Step 4: Since D 2 G3 ⊆ D 2 Gα for α ≥ 3, it follows that the space of identities
of D 2 Gα is contained in the space of identities for D 2 G3. We verify that the three
identities in degree 5 are satisfied by D 2 Gα over Q by independent computations
with the copular algebra on 20 alleles, as follows. Since each identity has degree
5 and is multilinear, and each natural basis element has 4 subscripts, we take
a = A1,2,3,4 , b = A5,6,7,8 , c = A9,10,11,12 , d = A13,14,15,16 , e = A17,18,19,20 ,
in the algebra D 2 G20. (We regard subscript j as short for the indeterminate
subscript ij.) An identity in degree 5 does not involve more than 20 alleles even
in an algebra on α > 20 alleles, so this is the general case. For α there is a
surjective homomorphism Φ from D 2 G20 to the subalgebra of D 2 Gα generated
by any 5 natural basis elements: if j ↦→ φ(j) is any function from {1, . . . , 20} to
{1, . . . , a} then Φ(Aj1j2j3j4) = A f(j1)f(j2)f(j3)f(j4). We now use arithmetic over Q
to evaluate the three identities in degree 5 on the elements a, b, c, d, e ∈ D 2 G20,
and find that each identity collapses to 0. It follows that the identities of the
Theorem are satisfied over Q by D 2 Gα for α ≥ 3, and hence the identities for
D 2 G3 are the same as the identities for D 2 Gα for α ≥ 3. ⊓⊔
Definition 43. We define polynomials U, V ∈ P5:
U = ((ab)c)(de) − ((ab)d)(ce) − ((ac)b)(de) + ((ac)d)(be) + ((ad)b)(ce)
− ((ad)c)(be) ,
V = 3((ab)c)(de) + 2((ac)b)(de) − 2((ac)e)(bd) + ((ad)b)(ce) − 4((ad)e)(bc)
− 3((bc)a)(de) + 3((cd)b)(ae) .
Theorem 44. Every identity of degree ≤ 7 satisfied by the copular algebra
D 2 Gα for α = 2 is a consequence of commutativity, R(a, b, c, d)e = 0, U ≡ 0
and V ≡ 0. The identities U and V are not satisfied by D 2 Gα for α ≥ 3.
Proof. The proof of the first claim is similar to that of Theorem 42. (For two
alleles, the space of identities in degree 5 has dimension 65.) For the second
claim, since D 2 G3 ⊆ D 2 Gα for α ≥ 3, it suffices to show that U and V are
not satisfied by D 2 G3. For a = A1111, b = A1111, c = A1112, d = A1113 and
e = A1112, both U(a, b, c, d, e) and V (a, b, c, d, e) are equal to
15A1122 + 86A1123 + 86A1212 + 15A1213 + 15A1223 + 86A1322 .
Since the result is nonzero over IFp, it is also nonzero over Q. ⊓⊔
Corollary 45. The following element is an absolute zero-divisor in the copular
algebra D 2 Gα over Q for any α ≥ 2 and any i1, . . . , i16 ∈ {1, . . . , α}:
R(Ai1i2i3i4, Ai5i6i7i8, Ai9i10i11i12, Ai13i14i15i16) =

14 Bremner, Piao and Richards
− 1
64
4
8
14
16
p=1 q=5 r=13 s=15
+ 1
256
4
8 12
p=1 q=5 r=9 s=13
Aipiqiris − 1
128
16
8
Aipiqiris + 1
128
16
10
12
p=1 q=13 r=9 s=11
12
14
8
Aipiqiris
16
p=1 q=9 r=13 s=15
Aipiqiris .
Proof. The identity R(a, b, c, d)e ≡ 0 says that the values of the recombination
identity are absolute zero-divisors in every copular algebra D 2 Gα. ⊓⊔
Example 46. The algebra D 2 G2 has dimension 6 and natural basis A1111, A1112,
A1122, A1212, A1222, A2222. For each of the 6 4 assignments in lexicographical
order of these basis elements to a, b, c, d we compute the element R(a, b, c, d).
We record the assignments which produce a result linearly independent of the
previous results. We obtain a subspace of absolute zero-divisors with basis
R(A1111, A1111, A1112, A1112) = 1
16 A1111 − 1
8 A1112 + 1
16 A1122 ,
R(A1111, A1111, A1112, A1212) =
1
16A1111 − 1
16A1112 + 1
16A1122 − 1
8A1212 + 1
16A1222 ,
R(A1111, A1112, A1112, A1212) =
References
5
128A1111 − 1
32A1112 + 3
64A1122 − 3
32A1212 + 1
32A1222 + 1
128A2222 .
1. Abraham, V. M.: Linearizing quadratic transformations in genetic algebras. Proc.
London Math. Soc. 40 (1980) 346–363.
2. Bernad, J., González, S., Martínez, C., Iltyakov, A. V.: On identities of baric
algebras and superalgebras. J. Algebra 197 (1997) 385–408.
3. Bernad, J., González, S., Martínez, C., Iltyakov, A. V.: Polynomial identities of
Bernstein algebras of small dimension. J. Algebra 207 (1998) 664–681.
4. Bernstein, S.: Solution of a mathematical problem connected with the theory of
heredity. Ann. Math. Statistics 13 (1942) 53–61.
5. Bremner, M. R.: Varieties of anticommutative n-ary algebras. J. Algebra 191 (1997)
76–88.
6. Bremner, M. R.: Jordan algebras arising from intermolecular recombination.
SIGSAM Bull. 39 (2005) 106-117.
7. Correa, I., Hentzel, I. R., Peresi, L. A.: Minimal identities of Bernstein algebras.
Algebras Groups Geom. 11 (1994) 181–199.
8. Etherington, I. M. H.: Genetic algebras. Proc. Royal Soc. Edinburgh 59 (1939)
242–258.
9. Etherington, I. M. H.: Nonassociative algebra and the symbolism of genetics. Proc.
Royal Soc. Edinburgh 61 (1941) 24–42.
10. González, S., Vicente, P., Mártinez, C.: Power-associative and Jordan second-order
Bernstein algebras. Nova J. Algebra Geom. 2 (1993) 367–381.
11. Hardy, G. H.: Mendelian proportions in a mixed population. Science 28 (1908)
49–50.

Polynomial Identities for Bernstein Algebras 15
12. Hentzel, I. R., Jacobs, D. P., Peresi, L. A., Sverchkov, S. R.: Solvability of the
ideal of all weight zero elements in Bernstein algebras. Comm. Algebra 22 (1994)
3265–3275.
13. Holgate, P.: Jordan algebras arising in population genetics. Proc. Edinburgh Math.
Soc. 15 (1966/1967) 291–294.
14. Holgate, P.: Genetic algebras satisfying Bernstein’s stationarity principle. J. London
Math. Soc. 9 (1974/75) 612–623.
15. Labra, A., Mallol, C., Suazo, A.: A characterization of power-associative Bernstein
algebras of order 2. Nova J. Algebra Geom. 3 (1994) 83–96.
16. Landweber, L., Kari, L.: The evolution of cellular computing: nature’s solution to
a computational problem. Biosystems 52 (1999) 3–13.
17. López-Díaz, M., Shestakov, I. P., Sverchkov, S. R.: On speciality of Bernstein
Jordan algebras. Comm. Algebra 28 (2000) 4375–4387.
18. Lyubich, Y. I.: Basic concepts and theorems of the evolutionary genetics of free
populations. Russian Math. Surveys 26 (1971) 51–124.
19. Lyubich, Y. I.: Mathematical Structures in Population Genetics. Springer-Verlag,
Berlin, 1992.
20. Martínez, C.: On nuclear Jordan-Bernstein algebras. J. Algebra 174 (1995) 453–
472.
21. Ouattara, M.: Sur les algèbres de Bernstein d’ordre 2. Linear Algebra Appl. 144
(1991) 29–38.
22. Peresi, L. A.: A basis of identities for Bernstein algebras. Int. J. Math. Game
Theory Algebra 14 (2004) 179–188.
23. Reed, M. L.: Algebraic structure of genetic inheritance. Bull. Amer. Math. Soc. 34
(1997) 107–130.
24. Sverchkov, S. R.: Structure and representations of Jordan algebras arising from
intermolecular recombination. Jordan Theory Preprint Archives, 25 May 2007:
homepage.uibk.ac.at/~c70202/jordan/archive/IR/index.html
25. Walcher, S.: Bernstein algebras which are Jordan algebras. Arch. Math. 50 (1988)
218–222.
26. Weinberg, W.: On the demonstration of heredity in man. in: Samuel H. Boyer
(editor), Papers on Human Genetics, Prentice-Hall, Englewood Cliffs, 1963.
27. Wörz-Busekros, A.: Algebras in Genetics. Springer-Verlag, Berlin, 1980.
28. Zhevlakov, K. A., Slinko, A. M., Shestakov, I. P., Shirshov, A. I.: Rings That Are
Nearly Associative. Academic Press, New York, 1982.