This paper defines new constructs to support aggregation in the temporal query language TQuel and presents their formal semantics in the tuple relational calculus. A formal semantics for Quel aggregates is defined in the process. Multiple aggregates; aggregates appearing in the where, when, and valid clauses; nested aggregation; and instantaneous, cumulative, moving window, and unique variants are supported. These aggregates provide a rich set of statistical functions that range over time, while requiring minimal additions to TQuel and its semantics. We show how the aggregates may be supported in an historical algebra, both in a batch and in an incremental fashion, demonstrating that implementation is straightforward and efficient.>

In this paper we discuss extensions to the conventional relational algebra to support transaction time. We show that these extensions are applicable to historical algebras that support valid time, yielding a temporal algebraic language. Since transaction time concerns the storage of information in the database, the notion of state is central. The extensions are formalized using denotational semantics. The additions preserve the useful properties of the conventional relational algebra.