In this chapter we introduce new closure conditions, much stronger than those considered before. These conditions apply primarily to classes of predicates but can be extended to classes of functions via graph predicates. They are nonfinitary, for they involve different forms of universal quantification. Our main interest is to find reflexive structures that are closed under this type of condition. While they are not total in the sense defined in Chapter 2, we shall see that they may satisfy important selector properties.
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