arXiv:1304.5421v1 [math.LO] 19 Apr 2013 Strongly representable atom structures Mohamed Khaled and Tarek Sayed Ahmed Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. Abstract An atom structure of type T is said to be strongly representable if all atomic algebras (of the same type T ) with that atom structure are representable. We show that for any finite n ≥ 3 and any signature T between Dfn and QEAn, the class of strongly representable atom struc- tures of type T is not elementary. We extensively use graphs and games as introduced in algebraic logic by Hirsch and Hodkinson. 1 Introduction In [3], Hirsch and Hodkinson proved that for finite n ≥ 3, the class of strongly representable cylindric-type atom structures of dimension n is not definable by any set of first-order sentences: it is not elementary class. Their method depends on that RCA n is a variety, an atomic algebra A will be in RCA n if all the equations defining RCA n are valid in A. From the point of view of AtA, each equation corresponds to a certain universal monadic second-order statement, where the universal quantifiers are restricted to ranging over the sets of atoms that are defined by elements of A. Such a statement will fail in A if AtA can be partitioned into finitely many A-definable sets with certain properties - they call this a bad partition. This idea can be used to show that RCA n (for n ≥ 3) is not finitely axiomatizable, by finding a sequence of atom structures, each having some sets that form a bad partition, but with the minimal number of sets in a bad partition increasing as we go along the sequence. This can yield algebras not in RCA n but with an ultraproduct that is in RCA n . In this article we extend the result of Hirsch and Hodkinson to any class of strongly representable atom structure having signature between the diagonal free atom structures and the quasi polyadic equality atom structures (recall the definitions of such algebras from [1] and [2]). As in [3] we deal only with finite dimensional algebras. Fix a finite dimension n<ω, with n ≥ 3. 2 Atom structures The action of the non-boolean operators in a completely additive atomic BAO is determined by their behavior over the atoms, and this in turn is encoded by the atom structure of the algebra. Definition 2.1. (Atom Structure) Let A = 〈A, +, −, 0, 1, Ω i : i ∈ I 〉 be an atomic boolean algebra with operators 1