topology of a topological field of sets can differ from the topology generated by taking arbitrary unions of complexes and in general the complexes of a topological field of sets need not be open or closed in the topology. Algebraic fields of sets and Stone fields A topological field of sets is called algebraic if and only if there is a base for its topology consisting of complexes. If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets are precisely the open complexes. Moreover, the open complexes form a base for the topology. Topological fields of sets that are separative, compact and algebraic are called Stone fields and provide a generalization of the Stone representation of Boolean algebras. Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the open elements of the interior algebra (which form a base for a topology). These complexes are then precisely the open complexes and the construction produces a Stone field representing the interior algebra - the Stone representation. (The topology of the Stone representation is also known as the McKinsey–Tarski Stone topology after the mathematicians who first generalized Stone's result for Boolean algebras to interior algebras and should not be confused with the Stone topology of the underlying Boolean algebra of the interior algebra which will be a finer topology). Preorder fields A preorder field is a triple where is a preordered set and is a field of sets. Like the topological fields of sets, preorder fields play an important role in the representation theory of interior algebras. Every interior algebra can be represented as a preorder field with its interior and closure operators corresponding to those of the Alexandrov topology induced by the preorder. In other words, for all : and Similarly to topological fields of sets, preorder fields arise naturally in modal logic where the points represent the possible worlds in the Kripke semantics of a theory in the modal logic S4, the preorder represents the accessibility relation on these possible worlds in this semantics, and the complexes represent sets of possible worlds in which individual sentences in the theory hold, providing a representation of the Lindenbaum–Tarski algebra of the theory. They are a special