On Supernilpotent Algebras

0.22 MB·

by  Alexander Wires

ON SUPERNILPOTENT ALGEBRAS
7 ALEXANDER WIRES
1
0
2 Abstract. We establish a characterization of supernilpotent Mal’cev algebras which
generalizestheaffinestructureofabelianMal’cevalgebrasandtherecentcharacterization
n
a of 3-supernilpotent Mal’cev algebras. We then show that for varieties in which the two-
J generated free algebra is finite: (1) neutrality of the higher commutators is equivalent

1 to congruence meet-semidistributivity, and (2) the class of varieties which interpret a
3 Mal’cev term in every supernilpotent algebra is equivalent to the existence of a weak

difference term.
]
A
R
.
h 1. Introduction
t
a
m
The notion of centralizing elements and centralizing subgroups plays a fundamental role
[
in the structure theory of groups, both finite and infinite. One of the most successful

1
approaches in extending aspects of these ideas to more general algebraic systems has
v
9 been the generalization of the commutator as a binary operation on congruence lattices
4
of algebras. The commutator theory for congruence permutable varieties developed by
9
8 J.D.H Smith [15], and latter extended to congruence modular varieties in Herrmann [9],
0 HagemannandHerrmann[8]andGumm[7], satisfies manyofthesameusefulpropertiesas
.
1 the traditional commutator when specialized to normal subgroups in the theory of groups.
0
In the setting of congruence modular varieties, the commutator has become a powerful
7
1 tool in understanding the structure of these algebras.
: The commutator, also known as the term-condition commutator, can be defined for
v
i arbitary algebras, and while it is no longer as well-behaved as in the congruence modular
X
setting, it has proven instrumental in numerous advances; for just two examples, in the
r
a classification of finite algebras by tame congruence theory [6] and then in the extensions of
many of these results to an infinite setting in Kearnes and Szendrei [10] and Kearnes and
Kiss [11]. This introduction can in now way give an accurate account and due recognition
to all the contributors to the general commutator theory - please consult Freese and
McKenzie [5].
Underlying many of the most impressive applications of the commutator for general
algebras is the fact that it is defined by a certain centralizer relation universally quantified
over the polynomials of an algebra which in principle carries information about the identi-
ties and behavior of the term operations in relation to particular elements. A fundamental
example of this is illustrated by the result in [9] which establishes the affine structure of
Date: January 30, 2017.
1
2 ALEXANDERWIRES
polynomials of an abelian algebra in a congruence modular variety; that is, an algebra in
a congruence modular variety is abelian if and only if it is polynomially equivalent to an
abelian group with a ring of operators.
The higher commutators defined by Bulatov [3] were introduced as a tool to help dif-
ferentiate between distinct Mal’cev clones on a finite set. Like the binary term-condition
commutator, it is also defined for arbitrary algebras by a centralizer relation which in
general satisfies many, but not all of the properties of the binary instance; however, in
Aichinger and Mudrinski [1] we find that in congruence permutable varieties many of the
properties that made the binary commutator such a useful tool continue to hold for the
higher commutators. The 3-supernipotent Mal’cev algebras (abelian in the sense of the
ternary commutator) are characterized in Mudrinski [13] as polynomially equivalent to
groups with a special family of unary and binary operators. This is a proper generaliza-
tion of the affine structure of abelian Mal’cev algebras from [9]. In Oprˇsal [14], the higher
commutators for Mal’cev algebras are encoded by a certain subpower with application to
supernilpotent clones. Recently, in Moorhead [12] some of the fundamental properties of
thehighercommutatorforcongruencepermutablevarietiesareextended tothecongruence
modular setting.
The purpose of the present article is to provide a characterization of supernilpotent
Mal’cev algebras which generalizes the affine structure of abelian Mal’cev algebras and
the special class of expanded groups described in [13]. The argument utilizes two principle
ideas: the use of absorbing polynomials to mimic aspects of the commutator calculus in
groups and the representation of arbitrary polynomials in loops by absorbing polynomi-
als. In [1], we learn the utility of absorbing polynomials in proving many of the higher
commutator properties in Mal’cev algebras; in particular, absorbing polynomials provide
a convenient generating set of the higher commutators of congruences [1, Lem.6.9]. In
[13], we see how absorbing polynomials are used to represent the polynomials of certain
expanded groups and determine “distributivity” of smaller arity absorbing polynomials.
For an arbitrary variety, congruence semidistributivity is equivalent to the neutrality
of the binary term-condition commutator. The conjecture is that for congruence semidis-
tributive varieties, all the higher commutators are also neutral. This is only proven under
the assumption that the two-generated free algebra is finite (Theorem 4.5). Under this
finiteness assumption, we also show the the class of varieties in which the supernilpotent
algebras are polynomially equivalent to n-type loops (defined in Section 3) is equivalent
to the class of varieties with a weak difference term.
Insection 2, we statethe definitions andproperties ofthe higher centralizer and commu-
tator. We also record the characterizations of 3-supernilpotent Mal’cev algebras from [13].
In section 3, we state and prove Theorem 3.8 which is the characterization of arbitrary
supernilpotent Mal’cev algebras. We also show how to derive the characterizations for
abelian and 3-supernilpotent Mal’cev algebras from Theorem 3.8. Section 4 contains the
results related to congruence meet-semidistributivity and varieties with a weak difference
term.
ON SUPERNILPOTENT ALGEBRAS 3
2. All and Sundry
We record some facts which we will require in the next section. The definitions and
notations for algebras are standard and can be found in [2] and [4]. For definitions and
notation concerning the commutator consult [5].
The natural numbers are N = {1,2,3,...} so 0 is not included (I apologize to half the
readers). For n ∈ N, let [n] = {1,...,n} and [n](k) = {s ⊆ [n] : |s| = k}. A partition of
n ∈ N is a sum n = n +···+n with n ∈ N; in particular, each n ≥ 1. Given a map
1 k i i
f : An → A and a ∈ A, we write f = aˆ to mean f is the constant map which maps An to
a.
We sometimes use a convenient notation for evaluating operations on particular substi-
tutions. Suppose we have an operation f(x ,...,x ) : An → A and a tuple (a ,...,a ) ∈
1 n 1 m
Am. Then for any C ⊆ [m] and tuple (b ,...,b ) ∈ A|C|, f(a ,...,a ) [b ,...,b ]
1 |C| 1 m C 1 |C|
denotes the evaluation of f where the elements of (b ,...,b ) are substituted in to the
1 |C|
corresponding coordinates of (a ,...,a ) specified C. For example, we have
1 m
f(a ,a ,a ,a ,a ,a ,a ) [5,2,1] = f(a ,5,a ,2,a ,a ,1).
1 2 3 4 5 6 7 {2,4,7} 1 3 5 6
We will also sometimes write a term t(x¯ ) for a subset S ⊆ [n] to specify that the variables
S
in t are precisely x for i ∈ S.
i
A quasigroup is an algebra A = hA,·,\,/i satisfying the identities
x\(x·y) ≈ y (x·y)/y ≈ x
x·(x\y) ≈ y (x/y)·y ≈ x.
In a quasigroup we always have the additional identities (x/y)\x = y = x/(y/x). A loop
is a quasigroup which has a constant 0 satisfying 0·x ≈ x·0 ≈ 0. In a loop, we have the
additional identities
0\x ≈ x x\x ≈ 0
x/0 ≈ x x/x ≈ 0.
Note any quasigroup has a Mal’cev term given by m(x,y,z) := (x/(y\y))·(y\z).
ForafixedvarietyV inthesignatureσ, wesayB = B,σB,F isanexpanded V-algebra
if the reduct B,σB ∈ V and F is a set of operations.
(cid:10) (cid:11)
(cid:10) (cid:11)
2.1. Nilpotent Mal’cev Algebras. Let A be an algebra in a Mal’cev variety. If A is
nilpotent, thenAispolynomiallyequivalent toanexpandedloop[5, Ch.7]. Let usdescribe
in more detail how this is accomplished. Let m(xyz) denote a Mal’cev term for A and
suppose A is nilpotent of class n. Inductively define the lower central series by
(1]0 = 1, (1]n+1 = [1,(1]n]
4 ALEXANDERWIRES
and sequence of terms f ,g by
n n
f (x,y,z) := x
0
f (x,y,z) := m(y,m(y,x,m(f (x,y,z),y,z)),f (x,y,z))
n+1 n n
and
g (x,y,z) := z
0
g (x,y,z) := m(g (x,y,z),m(m(x,y,g (x,y,z)),z,y),y).
n+1 n n
The lower central series and the terms f ,g are related in A in the following manner [5,
n n
Lem.7.3]: for any n ∈ N
m(f (x,y,z),y,z) (1]n x (1]n m(z,y,g (z,y,x))
n n
f (m(x,y,z),y,z) (1]n x (1]n g (z,y,m(z,y,x)).
n n
Since A is nilpotent of class n, (1]n = 0 ; thus, for all b,c ∈ A the maps x 7→ m(x,b,c) and
A
x 7→ f (x,b,c) are inverses of each other, and the maps x 7→ m(c,b,x) and x 7→ g (c,b,x)
n n
are inverses of each other. It also follows that
f (y,x,x) ≈ y g (x,x,y) ≈ y
n n
f (x,y,x) ≈ y g (x,y,x) ≈ y
n n
in A. We can now define the polynomial loop reduct. Fix an element 0 ∈ A and define
x·y := m(x,0,y) x\y := g (x,0,y) x/y := f (x,0,y).
n n
Then A is polynomially equivalent to the loop LA = hA,·,\,/,Fi where we gather the
remaining operations into F. LA is nilpotent of the same degree as A. In the next section,
we shall see how supernilpotence of A is related to characterizations of which operations
can be used for F in order to generate PolA.
2.2. Higher Commutators. Letusnowrecallthedefinitionofthen-commutatordefined
by Bulatov [3].
Definition 2.1. Let A be an algebra, and α ,...,α ,β ∈ ConA. We say that α ,...,α
1 n 1 n
centralizes β modulo δ, and write C(α ,...,α ,β;δ) if for all polynomials f(x¯ ,...,x¯ ,y¯)
1 n 1 n
¯ ¯ ¯
and vectors a¯ ,b ,...,a¯ ,b ,c¯,d satisfying
1 1 n n
¯
(1) a¯ ≡ b for all i = 1,...,n,
i αi i
¯
(2) c¯≡ d, and
β
¯ ¯
(3) f(x¯ ,...,x¯ ,c¯) ≡ f(x¯ ,...,x¯ ,d) for every tuple (x¯ ,...,x¯ ) ∈ a¯ ,b × ···×
1 n δ 1 n 1 n 1 1
¯ ¯ ¯
a¯ ,b \ (b ,...,b ) ,
n n 1 n (cid:8) (cid:9)
¯ ¯ ¯ ¯ ¯
we have f(b ,...,b ,c¯) ≡ f(b ,...,b ,d).
(cid:8) 1 (cid:9) (cid:8)n δ (cid:9)1 n
In this case we write A (cid:15)C(α ,...,α ,β;δ). The following properties follow directly by
1 n
application of the definition of the centralizer relation.
Lemma 2.2. We have the following properties of the centralizer relation in an agebra A:
ON SUPERNILPOTENT ALGEBRAS 5
(1) A (cid:15) C(α ,...,α ;δ ) → C(α ,...,α ; δ )
i∈I 1 n t 1 n t∈I t
(2) If each α ≤ β , then A (cid:15) C(β ,...,β ;δ) → C(α ,...,α ;δ).
i i 1 n 1 n
V V
(3) For any permutation σ : [n−1] → [n−1],
A (cid:15) C(α ,...,α ,α ;δ) iff A (cid:15) C(α ,...,α ,α ;δ).
1 n−1 n σ(1) σ(n−1) n
(4) For γ ≤ α ∧···∧α ∧δ,
1 n
A (cid:15) C(α ,...,α ;δ) iff A/γ (cid:15) C(α /γ,...,α /γ,β/γ;δ/γ).
1 n 1 n
(5) For i < n, A (cid:15) C(α ,...,α ,α ,...,α ;δ) → C(α ,...,α ;δ).
1 i−1 i+1 n 1 n
(6) If α ∧(β ◦(δ ∧α )◦β) ⊆ δ, then for k < n we always have
n n
A (cid:15) C(α ,...,α ,β,α ,...,α ;δ);
1 k−1 k+1 n
if in addition α ∧(γ ◦(δ ∧α )◦γ) ⊆ δ, then we have
n n
A (cid:15) C(α ,...,α ,β ∨γ,α ,...,α ;δ).
1 k−1 k+1 n
Property (1) in Lemma 2.2 yields the following definition.
Definition 2.3. For an algebra A, we let [α ,...,α ] be the smallest congruence δ such
1 n
that A (cid:15) C(α ,...,α ;δ).
1 n
For any arbitrary algebra A we have the the following properties:
(HC1) [α ,...,α ] ≤ α ;
1 n 1≤i≤n i
(HC2) If each α ≤ β , then [α ,...,α ] ≤ [β ,...,β ];
i i 1 n 1 n
V
(HC3) [α ,...,α ] ≤ [α ,...,α ].
1 n 2 n
If A generates a Mal’cev variety, then the following hold [1]:
(HC4) For any permutation σ : [n] → [n], [α ,...,α ] = [α ,...,α ];
1 n σ(1) σ(n−1)
(HC5) [α ,...,α ] ≤ η if and only if A (cid:15) C(α ,...,α ;η);
1 n 1 n
(HC6) If η ≤ α ∧···∧α , then in A/η we have
1 n
[α /η,...,α /η] = ([α ,...,α ]∨η)/η;
1 n 1 n
(HC7) For any j ∈ [n] and collection {β } ⊆ ConA,
i i∈I
[α ,...,α , β ,α ,...,α ] = [α ,...,α ,β ,α ,...,α ];
1 j−1 i j+1 n 1 j−1 i j+1 n
i∈I i∈I
_ _
(HC8) For any 1 < i < n, [α ,...,α ,[α ,...,α ]] ≤ [α ,...,α ].
1 i−1 i n 1 n
A congruence α ∈ ConA is n-supernilpotent if [α,...,α] = 0 . The algebra A is said to
A
n
be n-supernilpotent if 1 is. The terminology is justified by the following observation: if
A
A generates a Mal’cev variety, then by repeated a|ppl{iczat}ions of (HC8) for any α ∈ ConA
we have
[α,...,α] ≥ [α,[α,...,α]] ≥ [α,[α,[α,...,α]]] ≥ ··· ≥ [α,[···[α,α]].
n n−1 n−2
| {z } | {z } | {z }
6 ALEXANDERWIRES
with n-1 nested binary commutators in the last term. So if A is n-supernilpotent, then A
is nilpotent of class n-1.
The characterization of 3-supernilpotent Mal’cev algebras we wish to generalize is the
following:
Theorem 2.4. ([13])SupposeAgeneratesaMal’cevvariety. Thefollowingareequivalent:
(1) A is 3-supernilpotent ([1 ,1 ,1 ] = 0 ).
A A A A
(2) A is polynomially equivalent to an expanded group V = hA,+,−,0,Fi such that
(a) F is the set of at most binary absorbing operations on V,
(b) Every absorbing binary polynomial is distributive with respect to + in both
arguments, and
(c) V is 2-nilpotent ([1 ,[1 ,1 ]] = 0 ).
V V V V
3. Supernilpotent Mal’cev Algebras
Definition 3.1. [1] Let f(x ,...,x ) ∈ Pol A and a ,...,a ,a ∈ A. We say f absorbs
1 n n 1 n
(a ,...,a ) to a if f(b ,...,b ) = a whenever some b = a .
1 n 1 n i i
If a ∈ A, we say f(x ,...,x ) is a-absorbing if f absorbs (a,...,a) to a. Let AbkA
1 n a
denote the set of k-ary a-absorbing polynomials of A.
Absorbing polynomials interact with the commutator operation on the congruence lat-
tice in ways which mimic the behavior of the commutator term [x,y] := xyx−1y−1 in
relation to a normal series in a group G. Note [x,y] ∈ Ab2 G.
id
Lemma 3.2. Let A be an algebra. If f ∈ Pol A absorbs (a ,...,a ) to a, then for any
k 1 k
partition n = n +···+n with k ≤ n and {αi : i ∈ [k],j ∈ [n ]} ⊆ ConA we have
1 k j i
f(a /[α1,...,α1 ],...,a /[αk,...,αk ]) ⊆ a/ [α1,...,α1 ],...,[αk,...,αk ]
1 1 n1 k 1 nk 1 n1 1 nk
where we set [α] = α; consequently, if A is n-supern(cid:2)ilpotent, then AbkA = {ˆ0} for(cid:3)k ≥ n.
0
Proof: Let b ∈ a /[αi,...,α1 ] for i ∈ [k]. Then for any tuple u¯ ∈ a¯ ,¯b × ··· ×
i i 1 nI 1 1
¯ ¯ ¯
a¯ ,b \ (b ,...,b ) we see that f(u¯,a ) = a = f(u¯,b ) since some coordinate
k−1 k−1 1 k−1 k k (cid:8) (cid:9)
of u¯ must be an a . This yields
i
(cid:8) (cid:9) (cid:8) (cid:9)
a = f(b ,...,b ,a ) [α1,...,α1 ],...,[αk,...,αk ] f(b ,...,b ,b ).
1 k−1 k 1 n1 1 nk 1 k−1 k
Now assume A is n-supernilpo(cid:2)tent. Note 0/1 = A and so f(cid:3)or any f ∈ AbkA, we have
A 0
(f(b ,...,b ),0) ∈ [1,...,1] ≤ [1,...,1] = 0 by the first part and (HC3). •
1 k
k n
Definition 3.3. An|ex{pzan}ded l|oo{pzA}= hA,·,\,/,0,Fi is said to be n-type if
(1) F is the set of at most (n-1)-ary absorbing operations of A;
(2) every f ∈ Abn−1A is distributive over “·” in each coordinate;
0
ON SUPERNILPOTENT ALGEBRAS 7
(3) ConA satisfies [1,...,1,[1,...,1]] = 0 for each 2 ≤ k ≤ n .
A 2
n−k k (cid:6) (cid:7)
From condition (3) ab|ov{ez, w}e s|ee{tzha}t an n-type loop must be nilpotent of class n-1 by
repeated application of (HC8). The next two lemmas will be used in the proof of Theorem
3.8.
Lemma 3.4. Let A = hA,·,\,/,0,Fibe a n-type expanded loop. Fix k < n−1. Suppose
we have
• a parameter 1 ≤ p ≤ k with a ,...,a ∈ A and polynomials g ∈ Abmi A with
p+1 k i 0
m ≥ n for 1 ≤ i ≤ p.
i i
• a partition n−1 = n +···+n with n ≥ 2;
1 k 1
• f ∈ AbkA;
0
Then f(g ,...,g ,a ,...,a ) = ˆ0.
1 p p+1 k
Proof: By Definition 3.3(3) we have [[1,...,1,[1,...,1]] = 0 since n ≥ 2. Repeated
1
n−n1 n1
applications of (HC8) and (HC4) yields
| {z } | {z }
[[1,...,1],...,[1,...,1]] = 0. (∗)
n1 nk
Lemma 3.2 implies each (g ,|0){∈z[1},...,1]|≤{[z1,.}..,1]. We always have (a ,0) ∈ 1 and
i i A
mi ni
(h,0) ∈ 1 for any a ∈ A,h ∈ Ab1A. Then another application of Lemma 3.2 and (∗)
A i |0 {z } | {z }
finishes the calculation. •
Lemma 3.5. Let L = A = hA,·,\,/,0,Fi be a n-type expanded loop. If f ∈ AbkL with
0
1 ≤ k ≤ n−1, then for any left-associated product m y we have that
i=1 i
m Q
f(x ,...,x , y ,x ,...,x )
1 i−1 i i+1 k
i=1
Y
is a product of absorbing polynomials in the x ’s and y ’s of arity p with k ≤ p ≤ n−1.
i j
Proof: The proof is by descent from n − 1 down to 1. By permuting the coordinates,
it is enough to establish the lemma for the first coordinate. For the base case, take
g ∈ Abn−1L. By condition 3.3(2), g distributes over · so writing x¯ = (x ,...,x ) we
0 2 n−1
have g( m y ,x¯) = m g(y ,x¯). Now assume the result for k +1. Take f ∈ AbkL and
i=1 i i=1 i 0
define
Q Q
f′(y ,y ,x ,...,x ) := f(y ,x ,...,x )\(f(y ,x ,...,x )\f(y +y ,x ,...,x )).
1 2 2 k 2 2 k 1 2 k 1 2 2 k
8 ALEXANDERWIRES
Then f′ ∈ Abk+1L. Solving for f we have
0
m m−1 m−1
f( y ,x¯) = f( y ,x¯) f(y ,x¯)·f′( y ,y ,x¯)
i i m i m
!
i=1 i=1 i=1
Y Y Y
= (···((f(y ,x¯)·s )·s )···)·s
1 2 3 m
where s = f(y ,x¯)·f′( j−1,y ,x¯). By the inductive hypothesis each f′( j−1,y ,x¯) is a
j j i=1 j i=1 j
product of absorbing polynomials of arity p with k+1 ≤ p ≤ n−1; therefore, each s , and
j
also f( m y ,x¯), is a pQroduct of absorbing polynomials of arity p with kQ≤ p ≤ n−1. •
i=1 i
The next definition is meant to give a convenient generalization of when an algebra is
Q
polynomially equivalent to a module over a ring with unity.
Definition 3.6. Let A be an algebra. We say A has a representation of degree n if the
there exists 0 ∈ A and a binary polynomial x · y := r(x,y) ∈ Pol A such that every
2
polynomial f(x ,...,x ) ∈ Pol A can be written as a left-associated product
1 k k
p
f(x ,...,x ) = r = ((···((c·r )·r )···)·r )·r
1 k i 1 2 p−1 p
i=0
Y
where
(1) r = c ∈ A;
0
(2) p = min{k,n};
(3) 0·x = x·0 = x for all x ∈ A;
(4) for 1 ≤ m ≤ p, each r = r (x ,...,x ) is a left-associated product
m m 1 k
r (x ,...,x ) = tm(x¯ )
m 1 k S S
S∈Y[k](m)
where tm(x¯ ) ∈ AbmA.
S S 0
Example 3.7. Let A be an abelian algebra with a Mal’cev term m(x,y,z) and fix 0 ∈ A.
Define x+y := m(x,0,y). For each f ∈ Pol A, define
k
r (x) := m(f(0,...,0,x,0,...,0),f(0,...,0),0)
i
where x is in the i-th coordinate of f. Then r ∈ Ab1A. Since A is an abelian Mal’cev al-
i 0
gebra, one shows that f(x ,...,x ) = f(0,...,0)+ k r (x ). So A has a representation
1 k i=1 i i
of degree 1.
Let R = Ab1A and consider the operation + inPduced on R. Then + is commutative,
0
associative and there is an additive inverse defined by −x := m(0,x,0). So hR,+,−,0i
is an abelian group. If we define a multiplication on R by composition ◦, then R =
hR,◦,+,−,0i is a ring with unity given by id(x).
If we define an action of R on A by r∗a = r(a) for r ∈ R, a ∈ A, then this determines a
module representation of R which we denote by M . The degree 1 representation proves
A
A is polynomially equivalent to the R-module M in which each f ∈ Pol A is represented
A k
as f(x ,...,x ) = c+ k r ·x .
1 k i=1 i i
P
ON SUPERNILPOTENT ALGEBRAS 9
Since the binary operation is not associative in general, the order of the parentheses
matter.
Theorem 3.8. Let A be an algebra in a Mal’cev variety. The following are equivalent:
(1) A is n-supernilpotent;
(2) A is nilpotent and has a representation of degree n-1;
(3) A is polynomially equivalent to an n-type loop.
Proof: Let m(x,y,z) be a Mal’cev term for A.
(1) ⇒ (2): Assume A is n-supernilpotent. From the discussion in Section 2, A is
nilpotent of class n-1 and A is polynomially equivalent to a loop L = hA,·,\,/,0,Fi
A
which is also nilpotent of class n-1. We will establish the representation by a form of
polynomial interpolation in LA. We take the loop multiplication “·” as the basic binary
product and note that condition 3.6(3) is immediately satisfied.
Take f ∈ Pol A. Set f(0,...,0) = c. Inductively, we shall define for 0 ≤ k ≤ m
m
polynomials r (x ,...,x ) such that
k 1 m
(((···((c·r )·r )···)·r )·r )(¯0) [x¯ ] = f(¯0) [x¯ ] (∗∗)
1 2 k−1 k S S S S
for each S ∈ [m]k where ¯0 = (0,...,0). Set r (x¯) := f(0,...,0) = c ∈ A. For i ∈ [m],
0
define
t (x) := c\f(¯0) [x].
i {i}
Thent (0) = c\f(¯0) [0] = c\f(¯0) = c\c = 0; thus, t ∈ Ab1A. Thensetr (x ,...,x ) :=
i {i} i 0 1 1 m
t (x ) a left-associated product. We then check (∗∗):
i∈[m] i i
Q (c·r(x ,...,x ))(¯0) [x ] = c·t (x ) = c·(c\f(¯0) [x ]) = f(¯0) [x ].
1 m {i} i i i {i} i {i} i
Inductively, suppose r (x ,...,x ) is defined for k < m and satisfies (∗∗). Now for
k 1 m
S ∈ [m]k+1, define
k
t (x¯ ) := ( r )(¯0) [x¯ ] \f(¯0) [x¯ ]
S S i S S S S
!
i=0
Y
Then for any i ∈ S,
k
t (x¯ ) [0] = ( r )(¯0) [x¯ ] \f(¯0) [x¯ ]
S S i i S\{i} S\{i} S\{i} S\{i}
!
i=0
Y
= f(¯0) [x¯ ]\f(¯0) [x¯ ] = 0
S\{i} S\{i} S\{i} S\{i}
using the inductive hypothesis. This shows t (x¯ ) ∈ Abk+1A. Then define the left-
S S 0
associated product r (x ,...,x ) := t (x¯ ) and check (∗∗): fix T ∈ [m]k+1
k+1 1 m S∈[m]k+1 S S
Q
10 ALEXANDERWIRES
and calculate
k+1 k
( r )(¯0) [x¯ ] = ( r )(¯0) [x¯ ] ·t (x¯ )
i T T i T T T T
!
i=0 i=0
Y Y
k k
= ( r )(¯0) [x¯ ] · ( r )(¯0) [x¯ ] \f(¯0) [x¯ ]
i T T i T T T T
! ! !
i=0 i=0
Y Y
= f(¯0) [x¯ ].
T T
Now for k = m, note r (¯0) [x¯ ] = r (x ,...,x ) when i ≤ m. So by (∗∗) we have for
i [m] [m] i 1 m
the left-associated product
m m−1 m−1
r (x ,...,x ) = r (x ,...,x ) · r (x ,...,x ) \f(x ,...,x )
i 1 m i 1 m i 1 m 1 m
! ! !
i=0 i=0 i=0
Y Y Y
= f(x ,...,x ).
1 m
Since A is n-supernilpotent, by Lemma 3.2 t (x¯ ) = ˆ0 for S ∈ [m]k with k ≥ n; thus,
S S
r (x ,...,x ) = ˆ0 for k ≥ n. This reduces the representation above to
k 1 m
p
f(x ,...,x ) = r (x ,...,x )
1 m i 1 m
i=0
Y
where p = min{m,n−1}. Altogether, we have established that A has a representation of
degree n-1.
(1) ⇒ (3): Assume A is n-supernilpotent. Again we have that A is polynomially
equivalent to a loop L = hA,·,\,/,0,Fi which is nilpotent of class n-1. Condition 3.3(3)
A
is established by using (HC8). To show 3.3(2), let f ∈ Abn−1A and define
0
f′(y,z,x ,...,x ) := (f(y ·z,x ,...,x )/f(z,x ,...,x ))/f(y,x ,...,x ).
2 n−1 2 n−1 2 n−1 2 n−1
Then f′ ∈ AbnA and so by Lemma 3.2, f′ = ˆ0; thus,
0
f(y ·z,x ,...,x ) = f(y,x ,...,x )·f(z,x ,...,x ).
2 n−1 2 n−1 2 n−1
Inductively, we have for a left associated product m y ,
i=1 i
m m−1
Q
f( y ,x ,...,x ) = f( y ,x ,...,x )·f(y ,x ,...,x )
i 2 n−1 i 2 n−1 m 2 n−1
i=1 i=1
Y Y
m
= f(y ,x ,...,x )
i 2 n−1
i=1
Y
which is again left-associated. The same argument works for each coordinate. This estab-
lishes 3.3(2). From the implication (1) ⇒ (2) above, we have a representation of degree
n-1 for A. This implies we can take F′ = n−1AbkA ⊆ F in order to generate PolA and
k=1 0
so condition 3.3(1) is satisfied.
S