Data Algebra

The world's data is broken into billions of incompatible pieces. Tables in one system, documents in another, then graphs, key-value stores, spreadsheets, time series, geographic layers, vector embeddings - each held in its own format, queried in its own language, and walled off from everything around it. For forty years the industry's answer has been to build a separate database, with a separate query language and a separate storage engine, for every shape data can take.

This book argues that the fragmentation was never necessary.

Beneath every one of those shapes lies a single mathematical structure, and there is an algebra that unifies them all. It is data algebra, and this book provides its most comprehensive definition and exposition to date.

At data algebra's foundation stands a tower comprising four nested levels - couplets, relations, clans, and hordes - together with roughly a dozen operations that lift, unchanged, from one level to the next. You learn an operation once, and you have it everywhere. This produces an algebra that is closed, complete, and proven: every operation returns the same kind of structure it consumes, so queries compose without restriction and without end.

The first half of the book builds and explains that mathematics carefully from the ground up - a rigorous but readable proof that an algebra resting on the humble couplet can represent all data, without exception. The second half of the book puts the mathematics to work.

It introduces JabrDB, the first database built entirely on data algebra, and JQL, its algebraic query language, in which the operators of the algebra are themselves the query verbs. Because the algebra subsumes every shape data can take, a single engine can wear many "personalities" - relational, graph, document, OLAP, key-value, cartographic, time-series - each a lens onto the same algebraic core rather than a separate product reinvented from scratch.

A central chapter takes SQL apart. It catalogs fifty-six distinct problems with SQL and its dialects - the ambiguity of NULL, endless dialect fragmentation, forced normalization, broken composability, injection vulnerability, the table as a straitjacket into which all data must be forced - and shows, point by point, how data algebra dissolves all but six of them. No NULLs. No dialects. No mandatory normalization. No subqueries. Filters that are themselves data, and queries that chain indefinitely.

The closing chapters look outward: to a software development kit that would let any programmer build on the algebra, to algebraic connectivity that reaches the data stores already in existence, and to the long road from a desktop proof of concept toward an infrastructure that could one day reunite the world's scattered data under a single mathematical roof.

Written for software developers, data professionals, and students of mathematics and computer science - and for anyone who has ever suspected that the database landscape is far more complicated than it needs to be - the book is deliberately practical. It pairs naturally with the JabrDB application, for those who wish to try it. The reader can hold the theory in one hand and the keyboard in the other, watching each algebraic expression decompose, step by step, into something they can run and see.

Building on the foundational work begun by Professor Gary Sherman, who created this field of mathematics and computing, Data Algebra is at once a mathematical proof and a working manual - a complete course, in a single volume, in the structure that underlies all data.