Approximate distributive laws and finite equational bases
for finite algebras in congruence-distributive varieties
Kirby A. Baker and Ju Wang
Abstract. For a congruence-distributive variety, Maltsev’s construction of principal congruence relations is shown to lead to approximate distributive laws in the lattice of equivalence relations on each member. As an application, in the case of a variety generated by
a finite algebra these approximate laws yield two known results: the boundedness of the
complexity of unary polynomials needed in Maltsev’s construction and the finite equational
basis theorem for such a variety if of finite type. An algorithmic version of the construction
is included.
1. Introduction
We present a calculus of equivalence relations that quantifies Maltsev’s construction of principal congruence relations ([18], Theorem 1.20) to show how, in a
congruence-distributive variety, distributive laws hold for equivalence relations after
they have been adjusted by unary polynomial functions of a certain nesting depth.
This theory illuminates two results. The first, due to the second author [20], is that
in a congruence-distributive variety generated by a finite algebra, the nesting depth
of the unary polynomials involved in Maltsev’s construction of principal congruence
relations can be bounded, a concept here called bounded Maltsev depth. The second
is the theorem, due to the first author [4], that a congruence-distributive variety
generated by a finite algebra of finite type has a finite equational basis. Several
proofs of this latter result are in the literature [4, 16, 19, 15, 9]; see also [8]. In [21]
it is shown how this theorem follows from the boundedness theorem, to produce an
explicit equational basis while appealing neither to Ramsey’s theorem as in [4] or
to the compactness theorem of first-order logic in some form [16, 19, 15, 9, 17, 8].
An improved version of this construction is included in algorithmic form.
For a history of the question of finite equational bases for finite algebras, see
[4, 6, 17]. For general terminology and standard concepts, see [18, 10, 13]. Clark,
Davey, Freese, and Jackson [11] have extended and studied the concept of bounded
Maltsev depth for a variety or quasivariety under the name finitely determined syntactic congruences; yet another name for the concept is having term-finite principal
Date: October 12, 2004.
1991 Mathematics Subject Classification: Primary: 08B10; Secondary: 08A30, 08B26.
Key words and phrases: congruence distributive, principal congruence, finite basis.
Work of the second author supported by Chinese National Technology Project 97-3.
1
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K. A. BAKER AND J. WANG
congruences [7]. The case of a variety generated by a finite lattice was studied in
[5] and in [11].
2. Approximate distributive laws for equivalence relations
By an operational (or basic) unary polynomial on an algebra A we mean any
polynomial function obtained by freezing all entries except one in a basic operation
of A. Let OA be the set of all operational unary polynomials on A. For α ∈ Eqv(A),
the lattice of equivalence relations on the universe A of A, let Oα be the equivalence
relation on A generated by α ∪ {hp(a), p(b)i : p ∈ OA , ha, bi ∈ α}. Thus O is an
operator on Eqv(A). We write OA for O when needed for clarity.
2.1. Observations.
(1) O is a complete join-endomorphism of Eqv(A). In particular, O is isotone,
and therefore
(2) O(α ∩ β) ⊆ Oα ∩ Oβ for α, β ∈ Eqv(A).
(3) The fixed points of O are the congruence relations of A.
k
(4) Cg α = ∪∞
k=0 O α, where Cg α denotes the smallest congruence relation on
A that contains α. (This is a variant of Maltsev’s construction of principal
congruence relations; see [18], Theorem 1.20, [12] and §4 below.)
(5) If p is a unary polynomial function on A obtained by freezing all entries
but one in a term of depth D, then ha, bi ∈ α implies hp(a), p(b)i ∈ OD α.
(Here “depth” is nesting depth in the construction of the term.)
(6) For an integer M ≥ 0, by a linear unary polynomial of depth M let us
mean a composition of M operational unary polynomials of A (where a
composition of no polynomials is the identity function). If some linear
unary polynomial of depth at most M takes a pair ha0 , a1 i to the pair
hb0 , b1 i in A, let us say that hb0 , b1 i is weakly projective to ha0 , a1 i in at
most M steps, written ha0 , a1 i →M hb0 , b1 i. Then OM α is the equivalence
relation on A generated by all pairs weakly projective to pairs in α in at
most M steps.
(7) If f : A → B is a surjective homomorphism, then f ◦ OA = OB ◦ f .
Here f (α) for α ∈ Eqv(A) means the smallest member of Eqv(B) containing all pairs hf (a1 ), f (a2 )i for ha1 , a2 i ∈ α.
2.2. Lemma. Let V be a congruence-distributive variety, let t0 , . . . , tn be its Jónsson
terms, and let D be the maximum of their depths. Then for A ∈ V and α, β1 , . . . βm ∈
Eqv(A) we have these variants of the distributive laws:
(i): α ∩ (β1 ∨ · · · ∨ βm ) ⊆ (OD α ∩ OD β1 ) ∨ · · · ∨ (OD α ∩ OD βm )
(ii): (α ∨ β1 ) ∩ · · · ∩ (α ∨ βm ) ⊆ O2DL(m) α ∨ (O2DL(m) β1 ∩ · · · ∩ O2DL(m) βm ),
where L(m) = dlog2 me.
Proof. For (i): Suppose ha, ci ∈ α ∩ (β1 ∨ · · · ∨ β` ). Then ha, ci ∈ α and also
a and c are connected by a sequence a = b0 , . . . , bm = c such that for each j =
1, . . . , m we have hbj−1 , bj i ∈ βk(j) for some k(j). Let sij = ti (a, bj , c) for each
relevant i, j. As in [2], Jónsson’s laws relating the ti give a “zig-zag” sequence a =
APPROXIMATE DISTRIBUTIVE LAWS
3
s10 , s11 , . . . , s1m =s2m , s2,m−1 , . . . , s20 =s30 , s31 , . . ., etc., with the second subscript
alternately increasing and decreasing, ending with c in the guise of sn−1,m if n is
even or of sn−1,0 if n is odd. Let us examine relations between adjacent terms si,j−1
and sij . Since these terms are obtained by evaluating the unary polynomial ti (a, ·, c)
at bj−1 and bj respectively, by Observation 2.1-5 we have hsi,j−1 , sij i ∈ OD βk(j) .
By the same observation applied to the unary polynomial ti (a, bj , ·) evaluated at c
and a, we have hsij , ai ∈ OD α (since ti (a, bj , a) = a); similarly hsi,j−1 , ai ∈ OD α,
so hsi,j−1 , sij i ∈ OD α. Via the zig-zag sequence, then, ha, ci ∈ (OD α ∩ OD β1 ) ∨
· · · ∨ (OD α ∩ OD βm ), as required.
For (ii): For m = 2, if α, β1 , β2 were actually congruence relations, by congruencedistributivity we would have the derivation
(α ∨ β1 ) ∩ (α ∨ β2 ) ⊆ [(α ∨ β1 ) ∩ α] ∨ [(α ∨ β1 ) ∩ β2 ] ⊆
α ∨ [(α ∩ β2 ) ∨ (β1 ∩ β2 )] ⊆ α ∨ (β1 ∩ β2 ).
Since α, β1 , β2 are not necessarily congruence relations, however, we invoke (i) twice
and use Observations 2.1-1 and 2.1-2 to distribute powers of O through meets and
joins:
(α ∨ β1 ) ∩ (α ∨ β2 )
⊆ (OD α ∨ OD β1 ) ∩ OD α ∨ (OD α ∨ OD β1 ) ∩ OD β2
⊆ OD α ∨ (O2D α ∩ O2D β2 ) ∨ (O2D β1 ∩ O2D β2 )
⊆ O2D α ∨ (O2D β1 ∩ O2D β2 ).
For general m, it suffices to check the case where m is a power of 2, which is
accomplished by using the case m = 2 recursively:
(α ∨ β1 ) ∩ · · · ∩ (α ∨ β2m )
h
i
⊆ O2DL(m) α ∨ (O2DL(m) β1 ∩ · · · ∩ O2DL(m) βm )
i
h
∩ O2DL(m) ∨ (O2DL(m) βm+1 ∩ · · · ∩ O2DL(m) β2m )
⊆ O2D+2DL(m) α ∨ (O2D+2DL(m) β1 ∩ · · · ∩ O2D+2DL(m) β2m )
= O2DL(2m) α ∨ (O2DL(2m) β1 ∩ · · · ∩ O2DL(2m) β2m ).
2.3. Definition. For α ∈ Eqv(A), O−1 α is the largest β ∈ Eqv(A) with Oβ ⊆ α,
or equivalently (in view of Observation 1), O−1 α = ∨{θ ∈ Eqv(A) : Oθ ⊆ α}.
Further, O−k α means O−1 (O−1 (. . . (O−1 (α)) . . . )) (k times),
2.4. Observations.
(1) O−1 is a complete meet-endomorphism of Eqv(A).
(2) The fixed points of O−1 are the congruence relations on A.
(3) O−k α = ∨{θ ∈ Eqv(A) : Ok θ ⊆ α}.
Here is another kind of approximate distributive law, to be applied below; its
virtue is that the exponent of O for α does not depend on m:
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K. A. BAKER AND J. WANG
2.5. Lemma. O−mD [(α ∨ β1 ) ∩ · · · ∩ (α ∨ βm )] ⊆ OD α ∨ (OmD β1 ∩ · · · ∩ OmD βm ).
Proof. For convenience, for k = 0, . . . , m write ρk = OkD β1 ∩ · · · ∩ OkD βk . (Thus
ρ0 = 1 ∈ Eqv(A) and ρm occurs in the statement of the Lemma.) Let θ be such
that OmD θ ⊆ (α ∨ β1 ) ∩ · · · ∩ (α ∨ βm ); we must show that θ ⊆ OD α ∨ (OmD β1 ∩
· · · ∩ OmD βm ). We first prove this fact:
2.6. Claim. OD α ∨ (O(k−1)D θ ∩ ρk−1 ) ⊆ OD α ∨ (OkD θ ∩ ρk ) for k = 1, . . . , m.
The proof of the claim depends on the equation and inclusions
O(k−1)D θ ∩ ρk−1 = (O(k−1)D θ ∩ ρk−1 ) ∩ (α ∨ βk )
h
i
⊆ OD (O(k−1)D θ ∩ ρk−1 ) ∩ OD α ∨
h
i
OD (O(k−1)D θ ∩ ρk−1 ) ∩ OD βk
⊆ OD α ∨ OkD θ ∩ ρk
(2.7)
Here the equality follows from O(k−1)D θ ∩ ρk−1 ⊆ OmD θ ⊆ α ∨ βk . The first
inclusion follows from (i) of Lemma 2.2. In the second inclusion, it is harmless to
delete all terms but OD α in the bracketed expression on the left; for the bracketed
expression on the right, Observation 2 is used to distribute OD to the lowest-level
constituents, even within ρk−1 . To complete the proof of Claim 2.6, it suffices to
take the join of OD α with the first and last expressions in (2.7).
Now, since ρ0 = 1, by the claim we have θ = θ ∩ 1 = θ ∩ ρ0 ⊆ OD α ∨ (θ ∩ ρ0 ) ⊆
D
O α ∨ (θ ∩ ρ1 ) ⊆ · · · ⊆ OD α ∨ (θ ∩ ρm ) ⊆ OD α ∨ ρm , which gives the Lemma.
3. Varieties of bounded Maltsev depth
3.1. Definition. An algebra A has Maltsev depth at most M if OM +1 = OM on
Eqv(A), in which case Cg α = OM α for each α ∈ Eqv(A). A variety V can be said
to have Maltsev depth at most M if OM +1 = OM on Eqv(A) for all A ∈ V . An
algebra or variety has bounded Maltsev depth if it has Maltsev depth at most M for
some M .
3.2. Theorem (improving [21]). Let V be a congruence-distributive locally finite
variety. If the finite subdirectly irreducible members of V have bounded Maltsev
depth, then so does V itself. Specifically, if the Maltsev depths of finite subdirectly
irreducible members are bounded by N , then all members of V have Maltsev depth
bounded by N + D, where D is the maximum depth of the designated Jónsson terms
for V .
The proof appears in §4 below. The explicit bound on the Maltsev depth of V
is the improvement to [21].
3.3. Corollary ([20]). If V is a congruence-distributive variety generated by a finite
algebra then there is a bound M such that for all A ∈ V and all a, b ∈ A, Cg(a, b) =
CgM (a, b). Here Cg(a, b) denotes the principal congruence relation obtained by
APPROXIMATE DISTRIBUTIVE LAWS
5
identifying a and b and CgM (a, b) denotes the equivalence relation generated by
M
} (M -fold compositions).
{hp(a), p(b)i : p ∈ OA
Proof using Theorem 3.2. For A ∈ V and a, b ∈ A, apply Observation 2.1-4 to
δ(a, b), the atomic equivalence relation that identifies only a and b.
Note. In the terminology of [11], Corollary 3.3 asserts that a congruence variety
of finite type that is generated by a finite algebra has finitely determined syntactic
congruences.
3.4. Lemma. Suppose that f : A → B is a surjective homomorphism and that
B has Maltsev depth at most M . Then for each α ∈ Eqv(A) we have Cg α ⊆
OM α ∨ ker f .
In other words, in computing the congruence closure of an equivalence relation
we can bound powers of O in Eqv(A) at the cost of an adjustment by ker f .
k
∞
k
Proof. By various Observations, f (Cg α) = f (∪∞
k=0 O α) = ∪k=0 f (O α) =
∞
k
M
M
−1
M
∪k=0 O f (α) = O f (α) = f (O α). Therefore Cg α ⊆ f f (O α) =
OM α ∨ ker f .
3.5. Lemma. Let V be a congruence-distributive variety. If A ∈ V is the subdirect
product of finitely many factors each with Maltsev depth at most N , then A has
Maltsev depth at most N +D, where D is the maximum depth of designated Jónsson
terms for V .
Proof. Suppose A is a subdirect product of B1 , . . . , Bm , and let fi : A → Bi
be the corresponding coordinate projections. For α ∈ Eqv(A), Lemma 3.4 gives
Cg α ⊆ (ON α ∨ ker f1 ) ∩ · · · ∩ (ON α ∨ ker fm ). Then Cg α ⊆ OD ON α ∨ (ker f1 ∩
· · · ∩ ker fm ) = ON +D α ∨ 0 = ON +D α, by Lemma 2.5, taking into account the fact
that Cg α and the ker fi are congruence relations and so are fixed points of O and
O−1 .
Proof of Theorem 3.2. Let A ∈ V and α ∈ Eqv(A) be given. First consider the
case where α = δ(a, b). Let hr, si ∈ Cg(a, b). Because Maltsev’s construction (as
in Observation 2.1-4) is finitary, we still have hr, si ∈ Cg(a, b) inside some finitely
generated subalgebra S of A. Since V is locally finite, S is finite and is therefore
the subdirect product of finitely many factors. By Lemma 3.5 applied to S, we
have hr, si ∈ OM +D δ(a, b), as desired.
For general α, the same bound follows from Observation 1 and the fact that
every element of Eqv(A) is the supremum of a set of atomic equivalence relations,
i.e., relations of the form δ(a, b).
3.6. Examples. Consider varieties of lattices. Since only one nontrivial Jónsson
term is needed, such varieties have D = 2.
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K. A. BAKER AND J. WANG
(1) In the variety D of distributive lattices, the only subdirectly irreducible
member is the 2-element chain 2, which has Maltsev depth 0, so D has
Maltsev depth at most 0 + 2 = 2. This bound is actually achieved in the
cube 23 .
(2) The five-element nonmodular lattice N5 has Maltsev depth 2, so the variety
generated by N5 has Maltsev depth at most 4.
(3) The five-element modular nondistributive lattice M3 has Maltsev depth 2,
so the variety generated by M3 has Maltsev depth at most 4.
In the last two examples there are weak projectivities of length 3 that cannot be
shortened, but by using transitivities, Maltsev depth 2 is achieved.
4. An algorithmic approach
We present a direct proof of Lemma 3.5, by means of a recursive construction,
improving [20, 21]. It is based on the observation that when a Maltsev scheme
[12] is pulled back through a surjection, the weak projectivities pull back suitably
but equalities needed for the connecting sequence may fail, resulting in a longer
sequence that has “gaps”. The following framework provides for such gaps.
In any algebra A ∈ V, for a finite sequence of elements c = c0 ,. . . ,cm = d, by an
even link of the sequence (or an odd link) let us mean a pair hci , ci+1 i, where i is
even (or odd, respectively). For a, b ∈ A and an integer N , let us say that such a
sequence has depth (at most) N relative to a, b if
(i) m is odd, so that the number of terms is even; and
(ii) for all odd links hci , ci+1 i of the sequence, {a, b} →N {ci , ci+1 }, meaning that
either ha, bi →N hci , ci+1 i or either ha, bi →N hci+1 , ci i.
Let us call an even link hci , ci+1 i a gluing if ci = ci+1 or a gap if not. Let us say
that the sequence is end-consistent if
hc, ci i ∈ Cg(c, d) for all i.
Here are some examples, in all of which it is assumed that a, b, c, d ∈ A:
4.1. Examples.
(1) The trivial sequence: The sequence c, d itself has depth 0 relative to a, b,
since there are no odd links.
(2) A doubled sequence: If c = c0 , c1 , . . . , ck = d is a Maltsev sequence connecting c to d, with {a, b} →N {ci−1 , ci } for each i = 1, . . . , k, then the doubled
sequence c = c0 , c0 , c1 , c1 , . . . , ck , ck = d has depth N relative to a, b.
Conversely, observe that a sequence from c to d of depth N relative to
a, b that has no gaps, only gluings, gives a Maltsev sequence of depth at
most N witnessing hc, di ∈ Cg(a, b).
(3) An image sequence: If f : A → B is a homomorphism and c = c0 , c1 , . . . , cm =
d is a sequence in A of depth N relative to a, b, then c̄ = c̄0 , c̄1 , . . . , c̄m = d¯
is a sequence in B of depth N relative to ā, b̄, where bars denote images.
(4) A pullback sequence: If f : A → B is a surjection such that in B the
images c̄, d¯ are connected by a sequence of depth N relative to ā, b̄, then
this sequence pulls back to a sequence in A of depth N relative to a, b.
APPROXIMATE DISTRIBUTIVE LAWS
7
Indeed, the same terms can be used for the polynomials, with an arbitrary
choice of pre-images of the auxiliary elements involved.
(5) A lifted sequence: If c = c0 , . . . , cm = d is a sequence of depth N relative
to a, b, then the zig-zag sequence
c = c0 = t1 (c, c0 , d), t1 (c, c1 , d), . . . , t1 (c, cm , d) = t2 (c, cm , d), t2 (c, cm−1 , d),
. . . , t2 (c, c0 , d) = t3 (c, c0 , d), . . . , cm = d has depth N + D relative to a, b,
where D is the maximum depth of the terms ti .
Observe that the zig-zag sequence has the virtue of being end-consistent,
at the cost of an increase in depth. Observe also that if the original sequence
has no gaps, neither does the lifted sequence.
(6) A patched sequence: If c = c0 , . . . , cm = d is a sequence connecting c and
d and for some even i we have another sequence ci = r0 , r1 , . . . , rk = ci+1 ,
where k is even, then we say that the combined sequence c = c0 , . . . , ci , ci =
r0 , r1 , . . . , rk = ci+1 , ci+2 , . . . , cm has been obtained by “patching” the second sequence into the first at the link hci , ci+1 i. We generally re-index the
patched sequence.
Observe that if the original two sequences are end-consistent, so is the
patched sequence. Observe also that if the original two sequences have
depth at most N relative to a, b then so does the patched sequence.
Re-proof of Lemma 3.5. Given hc, di ∈ Cg(a, b), the plan is to start with the trivial
sequence c, d and modify it repeatedly by patching gaps, always keeping the result
end-consistent and of depth at most N + D relative to a, b, until finally such a
sequence is obtained with no gaps. Then we are done.
To describe the modification step, suppose that we currently have an endconsistent sequence c = c0 , . . . , cm = d of depth at most N + D relative to a, b. By
a “gap split by πj ”, where πj : A → Bj is the coordinate projection, let us mean a
gap hc0 , d0 i = hci , ci+1 i for which the images πj (ci ), πj (ci+1 ) are distinct—certainly
any gap has some such j. Patch this gap “via πj ” as follows. By end-consistency
in A, in Bj we have hπj (c0 ), πj (d0 )i ∈ Cg(πj (c), πj (d)) ⊆ Cg(πj (a), πj (b)). By
hypothesis, there is a Maltsev sequence in Bj connecting πj (c0 ) and πj (d0 ) with
depth at most N relative to πj (a), πj (b). Pull the double of this sequence back
to A and lift, to obtain an end-consistent sequence in A connecting c0 and d0 , of
depth at most N + D relative to a, b. Finally, patch this lifted sequence into the
current sequence at c0 , d0 to obtain a new sequence. By construction, the segment
of the new sequence between c0 and d0 has no gaps split by πj . Moreover, if later a
new end-consistent sequence is patched in somewhere in that segment, the resulting
patched sequence too will have no gaps split by πj between c0 and d0 , because by
end-consistency all the even links will be in ker πj .
A convenient overall organization is to patch all gaps split by π1 , via π1 , resulting
in a sequence with no gaps split by π1 ; then to patch all gaps split by π2 , via π2 ,
resulting in a sequence with no gaps split by π1 or π2 , and so on. The end result
is a sequence with no gaps split by any πi , or in other words, a sequence with no
gaps at all, as desired.
8
K. A. BAKER AND J. WANG
5. The finite basis theorem
5.1. Theorem ([4]). A congruence-distributive variety of finite type that is generated by a finite algebra is finitely based.
The proof appears later in this section.
5.2. Lemma. A variety V of finite type has bounded Maltsev depth M if and only
if this property is finitely equationally expressible, in the sense that there is a finite
set Σ of laws of V all of whose models have Maltsev depth at most M .
Proof. The “if” implication is trivial; let us consider “only if”. Because we are
constructing laws, this discussion will distinguish between three contexts: term
algebras, free algebras in V , and arbitrary algebras in V . We notate elements of
free algebras as images of terms. Thus for a term algebra T generated by variable
symbols x, y, . . ., the free algebra in V with corresponding generators will be denoted
FV (x̄, ȳ, . . . ), where the bar denotes images under the natural epimorphism of T
onto the free algebra. The proof of the lemma will consist of examining carefully
how the relation OM +1 = OM in a suitable free algebra becomes equational in V .
By a protolinear term, let us mean a term `(x, z1 , . . . , zm ) that is a formal composition of operation symbols using variable symbols x, z1 , . . . , zm , each appearing
once, where the auxiliary variable symbols z1 , z2 , . . . appear consecutively left to
right up to some point and do not appear thereafter, and where every subterm is
either a variable or includes x. For example, if the type consists of a single binary operation with symbol b and if m ≥ 2, then one protolinear term is `(x, z1 , . . . , zm ) =
b(z1 , b(x, z2 )). In an algebra in V with some designated elements c1 , . . . , cm there
is a corresponding unary polynomial a 7→ `(a, c1 , . . . , cm ) = b(c1 , b(a, c2 )). In fact,
every linear unary polynomial in every algebra in V has this form, for a suitable m.
Let us choose m large enough that protolinear terms in x, z1 , . . . , zm are adequate
to induce any linear unary polynomial of depth at most M + 1 in any member of
V ; such a choice is m = (M + 1)(k − 1), where k is the maximum arity of operation
symbols in the type of V . Let ΛM +1 be the set of protolinear terms ` in x, z1 , . . . , zm
of depth M + 1. Since V is of finite type, ΛM +1 is finite.
Take any ` ∈ ΛM +1 . In the free algebra F = FV (x̄0 , x̄1 , z̄1 , . . . , z̄m ), observe
that hx̄0 , x̄1 i →M +1 h`(x̄0 , z̄1 , . . . , z̄m ), `(x̄1 , z̄1 , . . . , z̄m )i. Then by the choice of M ,
h`(x̄0 , z̄1 , . . . , z̄m ), `(x̄1 , z̄1 , . . . , z̄m )i ∈ OM δ(x̄0 , x̄1 ), which by Observation 2.1-6 is
the equivalence relation generated by all pairs hp(x), p(y)i for all linear p that are
compositions of at most M operational unary polynomials on F. Therefore in F
there is a finite sequence connecting `(x̄0 , z̄1 , . . . , z̄m ), `(x̄1 , z̄1 , . . . , z̄m ), of depth
links, giving
at most M relative to x̄0 , x̄1 and with no gaps. The gluings of even S
equations in a free algebra, constitute a set Σ` of laws of V . Let Σ = `∈ΛM +1 Σ` .
Then V |= Σ.
Further, if in some model A of Σ we have ha0 , a1 i →M +1 he0 , e1 i, then there exist
` ∈ ΛM +1 and constants c1 , . . . , cm such that ej = `(aj , c1 , . . . , cm ) for j = 0, 1.
The laws of Σ` then give a recipe for building a Maltsev scheme in A to show
he0 , e1 i ∈ OM δ(a0 , a1 ). Formally, we represent A as a homomorphic image of F,
pull the arrow back to F, regard it as a Maltsev scheme, replace it by a Maltsev
APPROXIMATE DISTRIBUTIVE LAWS
9
scheme of depth at most M using the laws of Σ` , and then map forward to A. By
the observation of 2.1-6, this argument proves that A has Maltsev depth at most
M.
5.3. Remark. The construction just presented yields explicit laws, individually
not complex but possibly numerous.
Proof of Theorem 5.1. Let A be a finite algebra of finite type, generating a congruence-distributive variety V . By Theorem 3.2, V has bounded Maltsev depth M ,
so that Lemma 5.2 applies. Let us build an equational basis for V by including
various finite sets of laws in turn to get smaller and smaller varieties, ending with
V . First, let us take the finite set Ψ of laws of Jónsson [14] satisfied by the chosen
Jónsson terms for V . Second, let us take the finite set Σ of laws constructed in
Lemma 5.2; the variety defined by Ψ ∪ Σ includes V and has members of Maltsev
depth at most M .
Third, by Jónsson [14], V has only finitely many subdirectly irreducible (SI)
algebras, all finite; let K be their maximum cardinality. Let us take the set of
laws ∆K,M obtained by applying the construction of [2], §4, for the case of the
disjunction (∀x0 ) . . . (∀xK )(ORi<j xi ≈ xj ), to a maximum depth M . The set of
laws Ψ ∪ Σ ∪ ∆K,M defines a variety containing V of which all SI members have at
most K elements.
Fourth, a finite set Γ of additional laws will suffice to exclude the finitely many
SI models of Ψ ∪ Σ ∪ ∆K,M that are not in V . The equational basis of V , then, is
Ψ ∪ Σ ∪ ∆K,M ∪ Γ.
6. Problems
(1) Determine whether Lemma 3.5 can be extended to the case of infinitely
many subdirect factors. This is unlikely to be the case, even for lattices,
but a counterexample is elusive. One approach would be to look for a
sequence of finite lattices, each with OM +1 = OM for the same bound M ,
but where the Maltsev schemes producing this reduction require longer and
longer strings of transitivities.
(2) Can the approximate distributive law 2.5 be simplified while still retaining
a constant bound on the power of O applied to α? What about the case of
lattices? Can the power of O in 2.2-(ii) be reduced?
(3) From each operation ∗i on pairs described in [2] we can define an operation
α ∗i β on equivalence relations, whose value is the equivalence closure of the
obvious set of pairs. Incorporate these operations in the theory developed
in §2. (Cf. [4, 19].)
(4) The method of constructing a basis used in §5 is still not very economical in terms of the number of laws produced. Find a more economical
approach—one that comes close to reproducing known equational bases in
small examples.
10
K. A. BAKER AND J. WANG
(5) The method of [4] was actually carried further, to a finite basis theorem for
varieties whose subdirectly irreducible members form an elementary class.
This approach is distilled especially well in Jónsson [15]. Can this more
general theory be tied to the methods of the present paper?
The authors are grateful to the referee and editor for valuable suggestions.
References
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APPROXIMATE DISTRIBUTIVE LAWS
11
(K. A. Baker) University of California, Box 951555, Los Angeles, CA 90095-1555, USA
E-mail address: [email protected]
(J. Wang) College of Mathematics and Computer Science, GuangXi Normal University, Guilin, 541005, China