case of the general modal frames which are fields of sets with an additional accessibility relation providing representations of modal algebras. Algebraic and canonical preorder fields A preorder field is called algebraic (or tight) if and only if it has a set of complexes which determines the preorder in the following manner: if and only if for every complex , implies . The preorder fields obtained from S4 theories are always algebraic, the complexes determining the preorder being the sets of possible worlds in which the sentences of the theory closed under necessity hold. A separative compact algebraic preorder field is said to be canonical. Given an interior algebra, by replacing the topology of its Stone representation with the corresponding canonical preorder (specialization preorder) we obtain a representation of the interior algebra as a canonical preorder field. By replacing the preorder by its corresponding Alexandrov topology we obtain an alternative representation of the interior algebra as a topological field of sets. (The topology of this "Alexandrov representation" is just the Alexandrov bi-coreflection of the topology of the Stone representation.) While representation of modal algebras by general modal frames is possible for any normal modal algebra, it is only in the case of interior algebras (which correspond to the modal logic S4) that the general modal frame corresponds to topological field of sets in this manner. Complex algebras and fields of sets on relational structures The representation of interior algebras by preorder fields can be generalized to a representation theorem for arbitrary (normal) Boolean algebras with operators. For this we consider structures where is a relational structure i.e. a set with an indexed family of relations defined on it, and is a field of sets. The complex algebra (or algebra of complexes) determined by a field of sets on a relational structure, is the Boolean algebra with operators where for all if is a relation of arity then is an operator of arity and for all This construction can be generalized to fields of sets on arbitrary algebraic structures having both operators and relations as operators can be viewed as a special case of relations. If is the whole power set of then is called a full complex algebra or power algebra. Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it is isomorphic to the complex algebra