ON THE RELATIONSHIP OF AP, RS AND CEP IN CONGRUENCE

ON THE RELATIONSHIP OF AP, RS AND CEP
IN CONGRUENCE MODULAR VARIETIES. II
Clifford Bergman and Ralph McKenzie
Abstract. Let V be a congruence distributive variety, or a congruence modular
variety whose free algebra on 2 generators is finite. If V is residually small and has
the amalgamation property, then it has the congruence extension property. Several
applications are presented.
In two previous papers [1] and [2], we considered the following question: if V
is a residually small variety with the amalgamation property, must V have the
congruence extension property? Our work established the following implications
for a congruence modular variety V :
(1) If V is 2–finite and has C2, then AP + RS =⇒ R.
(2) If V is 4–finite with C2 and R, then AP + RS =⇒ CEP.
(The terminology will be explained below.)
In this paper we supplement and extend these results. Assuming still that V is
congruence modular, we have:
(3) AP + RS =⇒ C2.
(4) If V has R, then AP + RS =⇒ CEP.
Combining these implications, we have that every congruence modular, 2–finite
variety satisfies AP + RS =⇒ CEP. Furthermore, every congruence distributive
variety (no finiteness assumption) satisfies AP + RS =⇒ CEP.
Our universal algebraic notation and terminology are standard. Good references
are [4] and [9].
Let V be a variety of algebras. We say that V
• has the amalgamation property (AP) if, for all A, B0 , B1 ∈ V and all embeddings fi : A → Bi , for i = 0, 1, there is C ∈ V and embeddings gi : Bi → C,
i = 0, 1, such that g0 ◦ f0 = g1 ◦ f1 ,
• is residually small (RS) if there is a cardinal κ such that every subdirectly
irreducible algebra in V has cardinality less than κ,
• has the congruence extension property (CEP) if, for all A ≤ B ∈ V, and
congruence α on A, there is ᾱ ∈ Con B such that ᾱ A = α,
• is n–finite, for a positive integer n, if every member of V generated by n
elements is finite.
1991 Mathematics Subject Classification. Primary 08B10, 08B25; Secondary 03C25, 20E06..
Research partially supported by the Iowa State University Sciences and Humanities Research
Institute and National Science Foundation Grants No. DMS-8600300 and DMS-8701643
1
The least non-trivial congruence on a subdirectly irreducible algebra is called
the monolith, and the least and greatest congruences on an algebra are denoted 0
and 1 respectively. Let B be a subalgebra of A0 × A1 . It is often convenient to
denote the elements of B as vertical pairs xx01 with xi ∈ Ai for i = 0, 1. Let α ∈
Con A0 . Then there is an induced congruence α0 on B given by xx01 α0 yy01 ⇐⇒
x0 α y0 . Similarly, for β ∈ Con A1 , there is a congruence β1 on B defined in an
analogous way. The particularly important congruences 00 and 01 are denoted η0
and η1 respectively. Given a congruence β on an algebra
D(β) for the
o
n D, we write
x
2
subalgebra of D with universe β. That is: D(β) =
:xβy .
y
The commutator is a binary operation, denoted [ , ], defined on the congruence lattice of an algebra. In a congruence modular variety V, the commutator has
some strong properties. It is additive, that is
α,
_
_
Θ =
[ α, θ ]
for A ∈ V, α ∈ Con A, and Θ ⊆ Con A,
θ∈Θ
symmetric and meet-dominated
[ α, β ] = [ β, α ] ⊆ α ∧ β
for A ∈ V and α, β ∈ Con A,
respects finite direct products
for α, β ∈ Con A and γ, δ ∈ Con B
[ α0 ∧ γ1 , β0 ∧ δ1 ] = [ α, β ]0 ∧ [ γ, δ ]1 in A × B,
and exhibits the following homomorphism property
if f : A → B is a surjective homomorphism and α, β ∈ Con B then
f −1 [ α, β ] = f −1 α, f −1 β ∨ ker f.
For a systematic development of the subject, we direct the reader to [6], especially
chapter 4.
There are three identities involving the commutator that we consider in this
paper. We say that an algebra A has:
C1 if
∀α, β ∈ Con A
α ∧ [ β, β ] = [ α ∧ β, β ]
C2 if
∀α, β ∈ Con A
[ α, β ] = α ∧ β ∧ [ 1, 1 ]
R
∀B ≤ A
[ 1A , 1A ] B = [ 1B , 1B ] .
if
We say that a class K has one of these properties if and only if every member of K
has the property. Finally for congruences α, β on an algebra A, we define (α : β)
to be the largest congruence θ such that [ β, θ ] ≤ α.
We collect some facts connecting these concepts in the following theorem.
2
Theorem 0. Let V be a congruence modular variety.
(1) V satisfies C1 if and only if, for every subdirectly irreducible algebra A of
V with monolith β, if θ = (0 : β) then [ θ, θ ] = 0.
(2) V satisfies C2 if and only if every subdirectly irreducible algebra A of V
satisfies C2, and this in turn is equivalent to A being either abelian or
prime.
(3) C2 =⇒ C1.
(4) If V is a residually small variety then it satisfies C1. Conversely, if V is
finitely generated and satisfies C1, then it is residually small.
(5) Every variety with CEP satisfies C2 + R.
(1) and (4) are from [5], (2) and (5) are from [7]. (A is prime iff α, β 6= 0 =⇒
[ α, β ] 6= 0). (3) follows easily from the definitions.
Our first objective is the following
Theorem 1. Let V be a congruence modular variety. If V has AP and RS, then
V has C2.
We require a pair of lemmas. The first of these is virtually identical to Theorem
10.9 of [6]. We include the proof for completeness.
Lemma 2. Let V be a congruence modular, residually small variety. If V fails
C2, then V contains a subdirectly irreducible algebra A with monolith β, and an
endomorphism f such that
(1) 0 = [ β, β ] < β = [ β, 1 ]
(2) f = f 2
(3) x β y ⇐⇒ f (x) = f (y) β y.
Proof. By Theorem 0, V satisfies C1 since it is residually small. If V fails C2, there
is a subdirectly irreducible algebra D which is neither abelian nor prime. Let γ be
the monolith of D and κ = (0 : γ). Then, since D is not prime, [ γ, γ ] = 0 and
since D is not abelian, but satisfies C1, γ ≤ κ < 1 and [ κ, κ ] = 0.
Let ∆ = ∆γ,κ be the congruence on D(γ) generated by
x
y
,
:xκy .
x
y
Observe that on D(γ), κi = (ηi : γi ), for i = 0, 1, since
[ δ, γi ] ≤ ηi =⇒ [ δ ∨ ηi , γi ] = [ δ, γi ] ∨ [ ηi , γi ] ≤ ηi =⇒
δ ≤ δ ∨ ηi ≤ κi
by the homomorphism property of the commutator. Furthermore the following
relationships hold among the congruences of D(γ):
ηi ≺ γi ≤ κi
γ0 = γ1 = η0 ∨ η1
∆ ∨ ηi = κ0 = κ1
∆ ∧ ηi = 0
3
for i = 0, 1.
The last of these follows from the fact that [ γ, κ ] = 0. See [6, theorem 4.9]. It
follows from these identities that γ0 /η0 & η1 /0 % κ0 /∆ & η0 /0 % γ1 /η1 , and
therefore, η0 and η1 are atoms of Con D(γ).
We claim that ∆ is a completely meet-irreducible congruence with (unique) cover
κ0 . Let λ > ∆ and λ 6= κ0 . From the computation above, κ0 covers ∆, so λ κ0 .
Since κ0 = (η0 : γ0 ) we have [ λ, η0 ∨ η1 ] = [ λ, γ0 ] η0 and therefore [ λ, η1 ] η0 .
It follows that λ ∧ η1 6= 0. But η1 is an atom, so λ ≥ η1 , thus λ ≥ η1 ∨ ∆ = κ0 as
desired.
Define A to be D(γ)/∆. Then A is subdirectly irreducible with monolith β =
κ0 /∆. Note that [ κ0 , κ0 ] = [ η0 ∨ ∆, η1 ∨ ∆ ] ≤ ∆, so [ β, β ] = 0 on A. Also A
is non-abelian. For if it were abelian, then in D(γ), [ η1 , 1 ] ≤ ∆ ∧ η1 = 0. But
then [ γ0 , 1 ] = [ η0 ∨ η1 , 1 ] ≤ η0 which contradicts the fact that (η0 : γ0 ) = κ0 < 1.
Therefore, in A, [ β, 1 ] = β, by C1,
verifying
(1) of the lemma.
We define f : A → A by f xy /∆ = xx /∆. As ∆ ≤ κ0 , this is well-defined.
That f is an endomorphism satisfying conditions (2) and (3) above is a straightforward verification. In order to continue, we need to recall some facts about our modular variety V.
There is a ternary term d in the language of V, called the difference term, with the
following properties:
(1) V |= d(x, x, y) = y
(2) For every B in V, abelian congruence θ on B, and b ∈ B, hb/θ, di is a
ternary group, denoted M (θ, b), and for each n-ary term function t, if
t(b1 , b2 , . . . , bn ) = c then t is a ternary group homomorphism from M (θ, b1 )×
· · · × M (θ, bn ) to M (θ, c).
For the appropriate definitions and proofs see [6, 5.5–5.8].
Lemma 3. Let A be an algebra satisfying the conclusions of Lemma 2. There are
automorphisms e0 and e1 of A(β) given by:
x
d(x, f x, y)
x
x
e0
=
and
e1
=
y
y
y
d(x, f y, y)
where d is the difference term for V.
Proof. Recall that for any x, y ∈ A, xy ∈ A(β) ⇐⇒ x β y ⇐⇒ f (x) = f (y) β y
by (3) of Lemma 2. Suppose e0 xy = e0 uv . Then y = v and therefore x β u, so
f (x) = f (u) and all six elements lie in the same ternary group, M (β, x). We have
x − f (x) + y = d(x, f (x), y) = d(u, f (u), v) = u − f (x) + y, so x = u. Thus e0 is
injective. Similarly, one can check that
d(x, y, f x)
−1 x
e0
=
.
y
y
That e0 is a homomorphism follows from the fact that every term, in particular
every basic operation, is a ternary group homomorphism on the M (β, x)–blocks. Proof of Theorem 1. Assume that V has AP and RS, but fails C2. We shall derive
a contradiction. Let A, β, e0 and e1 be as in Lemmas 2 and 3. Since V is residually
4
small, there is a maximal essential extension, E, of A in V (see [12]). Observe that
E is subdirectly irreducible. Call its monolith µ. Without loss of generality, we
may assume that A ⊆ E and µ A ⊇ β.
The automorphisms e0 and e1 of Lemma 3 are also embeddings of A(β) into E2 .
Let us also define e2 : A(β) → E2 to be the identity map. By the amalgamation
property (applied twice), there is an algebra Q in V and maps sj : E2 → Q, for
j = 0, 1, 2, such that s0 ◦ e0 = s1 ◦ e1 = s2 ◦ e2 . Furthermore, there is a map
r : E → E2 given by r(x) = xx . Then rA : A → A(β), and figure 1 commutes.
Q .....................................................................
.......
.......
......
......
I
......
6@
.....
s0
s1 @ s2
u..................
...
...
@
...
...
...
@
...
...
...
2
2
2
..
...
E
E
E
...
...
...
...
I
@
@
I
6
...
...
r
@
@
..
..
e
...
1
e2
e0 @
@
..
.
@
@ ..?
A(β)
E
@
I
@
@
@
A
figure 1
Since E is a maximal essential extension of A, it is an absolute retract in V, that
is, a retract of each of its V-extensions. Therefore there is a retraction u : Q → E
such that u ◦ s2 ◦ r = idE . Define
ρi = ker(u ◦ si ) ∈ Con E2
for i = 0, 1, 2.
Claim. For each i, if ρi 6= 0, then either
ρi ≥ η0 ∧ µ1 and ρi ∧ η1 = 0 or
ρi ≥ η1 ∧ µ0 and ρi ∧ η0 = 0.
Proof of claim. Let us write ρ for ρi in the proof of the claim. For any non-zero
congruence α on E, [ 1E , α ] ⊇ [ 1E , µ ] ⊇ [ 1A , β ] = β 6= 0A ; that is, E is centerless.
Therefore, by the homomorphism property, E2 is centerless. Thus
0E2 6= [ 1, ρ ] = [ η0 ∨ η1 , ρ ] = [ η0 , ρ ] ∨ [ η1 , ρ ]
so, say [ η0 , ρ ] 6= 0. Then η0 ∧ ρ 6= 0, implying ρ ≥ η0 ∧ µ1 (since η0 /0 % 1/η1 .)
Furthermore, ρ ∧ η1 = 0, for if not, then ρ ≥ η1 ∧ µ0 , so ρ ≥ (η0 ∧ µ1 ) ∨ (η1 ∧ µ0 ) =
µ0 ∧µ1 . But choose (a, b) ∈ β−0A . Recall µ ⊇ β and f (a) = f (b). Since d is a term,
5
it follows that d(a, f a, a) µ d(b, f b, b) and therefore ei aa ≡ ei bb (mod µ0 ∧ µ1 ),
hence modulo ρ as well. But then u◦si ◦ei aa = u◦si ◦ei bb , so by the commutativity
of fig. 1, a = u ◦ s2 ◦ r(a) = u ◦ s2 ◦ r(b) = b, which is a contradiction. Therefore we
must have ρ η1 ∧ µ0 , so ρ ∧ η1 = 0. This proves the claim.
We first apply the claim to ρ2 . Certainly, ρ2 6= 0, in fact, xy ρ2 zz , for
z = u ◦ s2 xy (since u ◦ s2 zz = u ◦ s2 ◦ r(z) = z.) Let us assume that ρ2 ≥ η0 ∧ µ1 ,
and we will derive a contradiction. The case ρ2 ≥ η1 ∧ µ0 will follow by symmetry.
We have [ 1, ρ2 ] = [ η0 ∨ η1 , ρ2 ] ≤ η0 ∨ (η1 ∧ ρ2 ) = η0 , so [ 1, ρ2 ∨ η0 ] ≤ η0 implying
ρ2 ≤ η0 since E is centerless. For any x, y ∈ E, xy ≡ r ◦ u ◦ s2 xy = zz (mod ρ2 )
for some z ∈ E, whence x = z. Thus xy ρ2 xx ρ2 yx0 and we conclude that
ρ2 = η0 .
Therefore
A(β)
η0
−1
= e−1
2 (ρ2 ) = ker (u ◦ s2 ◦ e2 ) = ker (u ◦ s0 ◦ e0 ) = e0 (ρ0 ) .
a,a)
Observe that for every a ∈ A, e0 faa = d(f a,f
= aa . Choose (a, b) ∈ β − 0A .
a
Then
fa
fb
fa
fb
a
b
η0
=⇒ u ◦ s0 ◦ e0
= u ◦ s0 ◦ e0
=⇒
ρ0
.
a
b
a
b
a
b
Now apply the claim to ρ0 . If ρ0 ≥ (η0 ∧ µ1 ) then
ρ0 ∧ η1 = 0. If ρ0 ≥ (η1 ∧ µ0 ) then
a
b
ρ0
a
a
ρ0
b
b
, contradicting
−1
η0 = e−1
0 (ρ0 ) ≥ e0 (η1 ∧ µ0 ) ≥ η1 ∧ β0
on A(β) which is false, proving the theorem.
We now turn to our second theorem. In [1, theorem 6] it was proved that if V is
congruence modular, 4-finite and satisfies C2 and R, then V satisfies AP + RS =⇒
CEP. Theorem 1 above, eliminates the need to assume C2. The proof below
does without the assumption of 4-finiteness. In addition, it provides a considerable
simplification of the previous argument. We require one lemma from that paper.
Lemma. Let V be a congruence modular variety satisfying AP and RS. Let A be
a subdirectly irreducible member of V, and assume A is an essential extension of
B0 × B1 . Then either B0 or B1 is trivial.
Theorem 4. Let V be congruence modular and satisfy R. If V has AP and RS,
then V has CEP.
Proof. By Theorem 1, V has C2. Let A, B ∈ V, with A ≤ B and θ a completely
meet-irreducible congruence on A. It suffices to show that θ extends to B. By R,
[ 1B , 1B ] A = [ 1A , 1A ]. Thus A/ [ 1, 1 ] can be embedded in B/ [ 1, 1 ]. Suppose
first that θ ≥ [ 1, 1 ] on A. Since B/ [ 1, 1 ] is abelian, it has CEP, so the congruence
θ/ [ 1, 1 ] (of A/ [ 1, 1 ]) extends to a congruence ψ/ [ 1, 1 ] on B/ [ 1, 1 ]. Then ψ ∈
Con B and ψ A = θ.
6
D
f¯
@
I
@ ḡ
@
@
B
S
@
I
f@
@
g
@
A
figure 2
So we may assume that θ [ 1, 1 ]. Let f be the embedding of A into B. Define
S to be (A/θ) × A and g : A → S by g(a) = a/θ
. We apply the amalgamation
a
¯
property to (A, B, S, f, g) yielding (D, f , ḡ) with D ∈ V (fig. 2.)
Let η0 and η1 denote the coordinate projection kernels on S. It suffices to find
γ in Con D such that ḡ−1 (γ) = η0 . By the assumptions on θ, A/θ is subdirectly
irreducible with monolith µ. Thus, there is a congruence µ0 on S covering η0 . The
commutator respects finite direct products, so [ 1S , 1S ] = [ 1, 1 ]0 ∧[ 1, 1 ]1 ≥ µ0 ∧η1
since A/θ is non-abelian. Therefore
for all β ∈ Con S
(∗)
β η0 ⇐⇒ β ≥ µ0 ∧ η1
Proof: Certainly, η0 ≥ β ≥ µ0 ∧ η1 implies µ0 ∧ η1 = µ0 ∧ η1 ∧ η0 = 0, which is false.
Conversely, if β η0 then β ∨ η0 ≥ µ0 . Therefore by additivity and C2,
β ≥ [ β ∨ η0 , β ∨ η1 ] = (β ∨ η0 ) ∧ (β ∨ η1 ) ∧ [ 1, 1 ] ≥ µ0 ∧ η1 .
−1
S
Now let γ be a maximal congruence on D such
T that ḡ (γ) ≤ η0 . Then γ is
completely meet-irreducible. For suppose γ = j∈J γj with γj > γ, all j. Then
ḡ−1 (γj ) η0 , so by (∗), ḡ−1 (γj ) ≥ µ0 ∧ η1 , for all j ∈ J, and therefore
ḡ−1 (γ) =
\
ḡ−1 (γj ) ≥ µ0 ∧ η1
j∈J
which implies
ḡ−1 (γ) η0 ,
which is a contradiction.
Therefore D/γ is subdirectly irreducible and ḡ−1 (γ) = η0 ∧ δ1 for some δ ∈
Con A, by modularity. Then S/ḡ−1 (γ) ∼
= A/θ × A/δ and we have an induced
embedding ḡ/γ : A/θ × A/δ → D/γ. Furthermore, by the maximality of γ, this
embedding is essential. Therefore, by the lemma mentioned above, either A/θ or
A/δ is trivial. But θ [ 1, 1 ], so we must have A/δ trivial, which means δ = 1A ,
and therefore ḡ−1 (γ) = η0 as desired. Corollary 5. Let V be a congruence distributive variety. If V has AP and RS,
then V has CEP.
7
Corollary 6. Let V be a congruence modular variety that is 2-finite. If V has AP
and RS, then V has CEP.1
Proof. If V is congruence distributive, the commutator reduces to intersection.
Thus V trivially has R and the result follows from Theorem 4. Suppose V is
congruence modular and 2-finite. By theorem 1, V has C2. By [2, theorem 8] V
has R. Therefore, by Theorem 4, V has CEP. It is well-known that an arbitrary variety has enough injectives if and only if it
has AP, RS, and CEP. By the above Corollaries, it follows that any congruence
modular, 2-finite variety has enough injectives if and only if it has AP and RS.
As applications we offer the following Corollaries.
Corollary 7. Let V be a variety of groups with AP and RS. Then V is abelian.
Proof. By Theorem 1, V satisfies C2. It suffices to show that every subdirectly
irreducible group in V is abelian. Let G be subdirectly irreducible, and let M be
the normal subgroup corresponding to the monolith. Choose any a ∈ M − {1}, and
let A be the subgroup generated by a. Then A is abelian and G is an essential
extension of A. Therefore, by [2, Theorem 7], G is abelian. Remarks. In [10, page 43] H. Neumann states the following theorem: If V is a
variety of groups with AP, then either every finite member of V is abelian, or else
V is the variety of all groups.
Observe that the argument in Corollary 7 generalizes to any congruence modular
variety V such that
(i) the free algebra on 2 generators is abelian or
(ii) the free algebra on one generator is non-trivial, abelian and has an idempotent element.
Corollary 8. The variety of all squags is not residually small.
Proof. A squag is a groupoid hS, ·i satisfying
x·x =x
x·y =y·x
x · (x · y) = y.
Every squag is a quasigroup (“squag” is short for “Steiner quasigroup”), so it follows
that the variety of squags is congruence modular.
It is implicit in Bruck [3, Theorem 2.1] that the variety of squags has AP. On
the other hand, Quackenbush [11, 7.6] showed that this variety does not have CEP.
It is easy to see that the variety is 2-finite, in fact the free squag on {x, y} has 3
elements: x, y, x·y. Therefore, by Corollary 6, the variety of squags is not residually
small. A variety V is directly representable if it is finitely generated and contains only
finitely many finite, directly indecomposable algebras.
1 Added in Proof: Recently, Keith Kearnes has shown that the assumption in Corollary 6 that
V be 2-finite can be dropped. Thus, any congruence modular variety with AP and RS has CEP.
8
Corollary 9.
(1) Let V be a directly representable variety with AP. Then V has CEP. In fact,
V is a varietal product A⊗D, in which A is abelian and D is a discriminator
variety.
(2) Let V be a congruence modular, semi-simple, finitely generated variety with
AP. Then V = A ⊗ D in which A is abelian and D is filtral.
Proof. By [8], every directly representable variety is congruence permutable (hence
modular), satisfies C2 and every subdirectly irreducible member is either simple or
abelian. In case (2), semi-simplicity implies C2. Now let V represent the variety
in either case. V is finitely generated, hence 2-finite. V has C2, hence C1, hence is
residually small (Theorem 0). Therefore, by Corollary 6, V has CEP.
Let A be the subvariety of V consisting of all abelian algebras, and let D be
the subvariety generated by all simple, non-abelian algebras. Then by [7, 7.2],
V = A ⊗ D, and D is a congruence distributive variety, semi-simple and with CEP.
Therefore D is filtral, proving (2). In case (1), D is congruence permutable as well,
so it is a discriminator variety. References
1. C. Bergman, On the relationship of AP, RS and CEP in congruence modular varieties, Alg.
Univ. 22 (1986), 164–171.
2. C. Bergman, Another consequence of AP in residually small, congruence modular varieties,
to appear in Houston J. Math..
3. R. H. Bruck, A Survey of Binary Systems, Springer–Verlag, New York, 1971.
4. S. Burris and H. Sankappanavar, A Course in Universal Algebra, Springer–Verlag, New York,
1981.
5. R. Freese and R. McKenzie, Residually small varieties with modular congruence lattices,
Trans. Amer. Math. Soc. 264 (1981), 419–430.
6. R. Freese and R. McKenzie, Commutator Theory for Congruence Modular Varieties, London
Mathematical Society Lecture Notes Series 125, Cambridge University Press, New York, 1987.
7. E. Kiss, Injectivity and related concepts in modular varieties I–II, Bull. Australian Math. Soc.
32 (1985), 35–53.
8. R. McKenzie, Narrowness implies uniformity, Alg. Univ. 15 (1982), 67–85.
9. R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I, Wadsworth
& Brooks/Cole, Monterey, 1987.
10. H. Neumann, Varieties of Groups, Springer–Verlag, Berlin, 1967.
11. R. W. Quackenbush, Varieties of Steiner loops and Steiner quasigroups, Canad. J. Math. 28,
1187–1198.
12. W. Taylor, Residually small varieties, Algebra Universalis 2 (1972), 33–53.
CB: Iowa State University, Ames, Iowa 50011
RM: University of California, Berkeley, CA 94720
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