A Mathematics of Data: Toward a Shape-Independent Algebra for Data Management

Relational algebra supplied data management with a rigorous mathematical foundation, but one whose object is a single data structure: the table. As document, graph, key-value and other non-tabular models have proliferated, each has been accompanied by its own model of operations and its own query language, with no common formal ground between them. This paper argues that the resulting fragmentation is a consequence of having a mathematics of tables rather than a mathematics of data. It outlines the Algebra of Data, developed by Sherman and Bloor and derived from Zermelo-Fraenkel set theory, which takes a single atomic unit, the couplet (an ordered pair), and constructs from it, by iterated power-set formation, a hierarchy of three further levels: the relation, the clan and the horde. A small set of operations (composition, transposition and the Boolean set operations) is defined at the level of couplets and relations and lifts, without redefinition, to the higher levels. We show how graphs, tables and nested documents are represented within this hierarchy without structural distortion, and how a single operation, composition, accounts for what are conventionally treated as distinct operations: the relational join, the graph traversal and the document path lookup. The paper is introductory; it states the framework and its motivation and defers full formal development to the cited works.