COLOURINGS OF HYPERGRAPHS, PERMUTATION GROUPS ...

4 B. LAROSE AND L. HADDAD
Acknowledgments.
The authors wish to thank Ágnes Szendrei and Keith Kearnes for fruitful discussions.
2. Preliminaries
Unless otherwise specified, the groups we consider in the sequel are concrete
groups of permutations. Recall that for G a group of permutations on A and x ∈ A,
the set of all σ(x) with σ ∈ G is called the orbit of x under G. The group G is
transitive if it has a single orbit, i.e. for every x, y ∈ A there exists σ ∈ G such that
σ(x) = y. For x ∈ A the stabiliser of x is the subgroup G x = {σ ∈ G : σ(x) = x};
for S ⊆ A the (set-wise) stabiliser of S is G S = {σ ∈ G : σ(S) ⊆ S}. A transitive
group G is regular if for each x and y in A the permutation taking x into y is
unique; equivalently, if every non-identity permutation in G is fixed-point free, i.e.
|G x | = 1 for all x ∈ A. A group G is primitive if there is no non-trivial equivalence
relation θ on the set A which is invariant under the permutations of G (viewed as
unary operations on A).
It will be convenient (and interesting) from time to time to consider our problem
from an algebraic point of view. For any group of permutations G on a set A let
A G denote the (non-indexed) algebra 〈A; P ol ρ G 〉. A set of operations on A which
contains all projections and is closed under composition is called a clone. It is easy
to verify that P ol ρ G is a clone; its members are called the terms of the algebra
A G . The unary part of a clone C is the set of its unary operations. It is a simple
exercise to verify the following fact: a clone C has unary part equal to G if and
only if it is contained in the clone P ol ρ G . In particular, the only unary operations
that preserve ρ G are the members of G, and hence the relational structure 〈A; ρ G 〉
is a core. Thus Theorem 1.1 above can be invoked to prove NP-hardness.
Since a Taylor operation preserving the relation ρ G is an idempotent term of the
algebra A G , we may use the following technique to produce “obstructions” to the
existence of such a term. A k-ary relation µ on A is an idempotent k-subalgebra of
A G if it is preserved by every idempotent term of A G . (In other words, these are
the subalgebras of the algebra B k where B is the idempotent reduct of A.) When
the arity of the relation is clear we shall simply say idempotent subalgebra.
Alternatively, these relations can be described as follows. A first-order formula
φ in the language of ρ G (with equality and constants) is primitive positive if it is
built up using only the existential quantifier and conjunction, i.e. of the form
φ ≡ ∃x 1 · · · ∃x m ψ
where ψ is a conjunction of atomic formulas involving ρ G , equality and the unary
relations {a} for every a ∈ A.
Lemma 2.1 ([1]). Let µ be a k-ary relation on A. Then µ is an idempotent
k-subalgebra if and only if there exists a primitive positive first-order formula
φ(x 1 , . . . , x k ) in the language of ρ G (with equality and constants) with free variables
x 1 , . . . , x k such that
µ = {(a 1 , . . . , a k ) : φ(a 1 , . . . , a k ) holds}.
Here are examples of such constructions. Let θ be a k-ary relation on A. For
each sequence of indices I = (i 1 , . . . , i j ) where 1 ≤ i s ≤ k for all s, we define the