COLOURINGS OF HYPERGRAPHS, PERMUTATION GROUPS ...

10 B. LAROSE AND L. HADDAD
noticing that D 4 consists of all affine transformations x ↦→ ax + b with a, b ∈ Z/4Z
and a invertible. It follows that the operation M(x, y, z) = x − y + z preserves ρ D4
(where the sum is in Z/4Z.)
□
We shall now prove that no Taylor operation preserves the relation ρ Dn for n ≠ 4.
For this we require the following results on graphs, one of which was mentioned
earlier in section 2.
Lemma 4.2 ([2], [30]). Let θ be a binary, areflexive, symmetric relation on a set
A and let Γ be the associated simple graph. If Γ is not bipartite then θ is invariant
under no Taylor operation.
Let θ be a binary, reflexive relation. We’ll say that θ is intransitive if whenever
(a, b), (b, c), (a, c) ∈ θ then |{a, b, c}| ≤ 2. We’ll say that θ is not acyclic if its
symmetric closure (with all loops removed) is not a tree.
Lemma 4.3 ([29]). Let θ be a binary, reflexive relation on A which is intransitive.
If θ is not acyclic then it is invariant under no Taylor operation.
Theorem 4.4. If n ≠ 4 then no Taylor operation preserves the relation ρ Dn ; in
particular the problem CSP (D n ) is NP-complete.
Proof. We divide the proof in 3 cases. Suppose first that n is odd: consider the
projection θ of ρ Dn onto the coordinates 0 and 1 (an edge of the n-gon). It is easy
to see that θ is symmetric, areflexive, and is in fact the n-cycle; by Lemma 4.2 we
are done.
Now suppose that n is even. Consider the relation α that consists of all pairs
(x 0 , y 0 ) such that there exist (x 0 , . . . , x n−1 ), (y 0 , . . . , y n−1 ) ∈ ρ Dn with x 1 = y 1 . It
is easy to see that
α = {(x, y) : x − y = ±2}
where the sum is taken modulo n. Thus α is a reflexive, symmetric relation. If n ≥ 8
then this is an intransitive cycle so we apply Lemma 4.3; if n = 6, then α is a congruence
of A D6 (on whose blocks D 6 acts regularly), and we define the idempotent
subalgebra γ as the set of all pairs (x 0 , x 2 ) such that there exists (x 0 , . . . , x 5 ) ∈ ρ D6
such that (x 0 , 0) ∈ θ. It is easy to see that γ is an areflexive, symmetric cycle of
length 3, and hence Lemma 4.2 applies.
□
4.2. Affine transformations. In this section we completely determine the complexity
of CSP (G) in the case where G is the group of all (bijective) affine transformations
on a matrix ring. 2 (See [27] for a study of monoidal intervals related to
affine transformations.) More precisely, let S be a (finite) commutative ring with
1, let n be a positive integer and let R = M n (S) the ring of n × n matrices over
S. Let U = GL n (S) denote the group of invertible matrices in M n (S). The group
G = G n,R will consist of all transformations on R of the form x ↦→ ax + b where
a ∈ U and b ∈ R. It is clear that G acts transitively on R.
Theorem 4.5. Let G = G n,R . Then the following conditions are equivalent:
(1) A G admits a Taylor term;
(2) n = 1 and the operation m(x, y, z) = x − y + z is a term of A G .
If one of these conditions holds then CSP (G) is in P; otherwise it is NP-complete.
2 For basic results on rings we refer the reader to [10] and [21].