  Association Schemes and coherent configurations (and the related Bose-Mesner
algebra and coherent algebras) are well known in combinatorics with many
applications. In the 1990s, Mesner and Bhattacharya introduced a
three-dimensional generalisation of association schemes which they called an
association scheme on triples (AST) and constructed examples of several
families of ASTs. Many of their examples used 2-transitive permutation groups:
the non-trivial ternary relations of the ASTs were sets of ordered triples of
pairwise distinct points of the underlying set left invariant by the group; and
the given permutation group was a subgroup of automorphisms of the AST. In this
paper, we consider ASTs that do not necessarily admit 2-transitive groups as
automorphism groups but instead a transitive cyclic subgroup of the symmetric
group acts as automorphisms. Such ASTs are called circulant ASTs and the
corresponding ternary relations are called circulant relations. We give a
complete characterisation of circulant ASTs in terms of AST-regular partitions
of the underlying set. We also show that a special type of circulant, that we
call a thin circulant, plays a key role in describing the structure of
circulant ASTs. We outline several open questions.
