COLOURINGS OF HYPERGRAPHS, PERMUTATION GROUPS ...

8 B. LAROSE AND L. HADDAD
3.3. Simple algebras. If we wish to determine whether there exists a Taylor term
that preserves the relation ρ G , it seems natural to try to break down the problem
into smaller parts, as for instance, if G can be decomposed as a wreath product of
smaller groups. In another vein, if G is not transitive on A, then we may consider
its actions on orbits: for each orbit A ′ ⊆ A, we have a group of permutations G ′
on A ′ consisting of the restrictions of the members of G to A ′ . In [14], Haddad
and Rödl ask whether there exists a group G such that for every group G ′ obtained
this way, the problem CSP (G ′ ) is in P, but CSP (G) is NP-complete; we shall
provide examples in the next section. Unfortunately, it seems unlikely that a characterisation
of the tractable cases can be obtained in the transitive case only, as
the following discussion will show.
Another approach to break down our problem is to use congruences of the algebra
A G , and analyse the quotients. We require the following result from [33] (see also
Lemma 3.5 of [26]):
Lemma 3.3. Let G be a transitive group of permutations on A and let A be a
simple algebra whose clone of term operations has unary part equal to G. Then
either G is regular or A is essentially unary.
Corollary 3.4. Let G be a transitive group of permutations on A. If the algebra
A G is simple, in particular if the group G is primitive, then A G has a Taylor term
if and only if the discriminator is a term of A G . If this holds CSP (G) is in P,
otherwise it is NP-complete.
Proof. If the algebra A G has a Taylor term then it is not essentially unary; the
result then follows immediately from Lemma 3.3 and Proposition 3.1. □
Suppose that G is transitive and that A G is not simple. If α is a maximal
congruence of A G then the quotient algebra A G /α is simple; furthermore the unary
part of its clone of term operations consists of the permutations of the blocks of α
induced by the elements of G; this is a transitive group, and thus by Lemma 3.3
either the quotient algebra is essentially unary, and hence A G has no Taylor term,
or else the action of G on the blocks of α is regular. Then the following is a term
operation of A G :
{
z, if (x, y) ∈ α,
T (x, y, z) =
x, otherwise.
Indeed, if σ, τ, µ ∈ G are such that (σ(x), τ(x)) ∈ α for some x ∈ A, then the same
holds for every x ∈ A by regularity on the blocks. Thus T (σ(x), τ(x), µ(x)) = µ(x)
for all x ∈ A. Otherwise we get T (σ(x), τ(x), µ(x)) = σ(x) for all x ∈ A and so T
preserves ρ G .
Hence we obtain the following:
Proposition 3.5. Let G be a transitive group of permutations on A. If A G has a
Taylor term then it has a quotient with a discriminator term.
Lemma 3.6. Let G be a transitive group of permutations on A, let α be a congruence
of A G such that G acts regularly on the blocks of α; let B be a block of α and
let H = G B . If the relation ρ G is invariant under a Taylor operation then so is
ρ H . If the relation ρ H is invariant under a Mal’tsev operation then so is ρ G .
Proof. As we remarked earlier, the fact that G acts regularly on the blocks of α
translates as follows: for any σ, τ ∈ G, if there exists some x ∈ A such that σ(x) and