* Relational Algebra Engine
  :PROPERTIES:
  :CATEGORY: relational-algebra
  :END:
  A pure Erlang implementation of E. F. Codd’s relational algebra.
  Relations are treated as mathematical sets of tuples; operations are defined exactly
  as in relational theory, with optional aleph₀ (infinite) relations.

** Core Specification
   The theoretical backbone of the system.

*** TODO Define relation as a strict mathematical set
    - [ ] Each tuple is a set of attributes (no duplicates, no ordering)
    - [ ] Each attribute = (name, value) pair
    - [ ] Attribute domains are strongly typed

*** TODO Enforce "Codd's axioms" in implementation
    - [ ] No duplicate tuples in any relation
    - [ ] No implicit ordering of tuples or attributes
    - [ ] All operations closed over the relation type
    - [ ] Proper domain checking in all operations

** Data Model Layer (`relation.erl`)
   Fundamental constructs of the relational model.

*** TODO Implement attributes, tuples, relations, and schemas
*** TODO Add schema validation for domain and arity
*** TODO Add conversion helpers:
    - [ ] tuple <-> map
    - [ ] schema validation checks

** Algebraic Operations (`operations.erl`)
   The complete set of classical relational algebra operators.

*** TODO Selection (σ)
*** TODO Projection (π)
*** TODO Cartesian Product (×)
*** TODO Natural Join (⋈)
*** TODO Theta Join (⋈θ)
*** TODO Union (∪)
*** TODO Intersection (∩)
*** TODO Difference (−)
*** TODO Rename (ρ)
*** TODO Set semantics enforcement (remove duplicates after each op)

** Infinite Relations (`aleph0.erl`)
   Representation of aleph₀-relations (countably infinite generators).

*** TODO Implement integer() generator
*** TODO Implement plus(), minus(), multiply(), neg() relations
*** TODO Add constraints and modes for partially bound arguments
*** TODO Implement lazy generation using streams or coroutines
*** TODO Support user-defined infinite relations

** Unified Relation Abstraction (`unified_relation.erl`)
   A bridge between finite and infinite relations.

*** TODO Implement unified type for finite and aleph₀
*** TODO Implement unified select, project, join, union, etc.
*** TODO Support type checking between finite and infinite relations
*** TODO Add automatic materialization with configurable limits
*** TODO Implement caching/memoization for aleph₀ evaluations

** Backend Integration (Mnesia)
   Persistent storage and retrieval of finite relations.

*** TODO Map finite relations to Mnesia tables
*** TODO Support schema registration and domain enforcement
*** TODO Add persistence for partially materialized aleph₀ relations
*** TODO Add atomic transactional updates
*** TODO Add indexing support for joins

** Correctness and Validation
   Ensuring theoretical fidelity.

*** TODO Implement relational algebra test suite
*** TODO Verify closure and idempotence of all operations
*** TODO Add property-based tests with PropEr or QuickCheck
*** TODO Compare semantics with formal definitions (e.g., Date & Darwen)

** Developer Tools
   Optional but desirable tools for exploration and testing.

*** TODO Add REPL-like console for interactive relational queries
*** TODO Add pretty-printer for relations and schemas
*** TODO Add visualization of join graphs and operation pipelines
*** TODO Add Org Babel integration for relational queries

** Documentation
   Literate and theoretical exposition.

*** TODO Write README with theoretical introduction
*** TODO Add module-level comments summarizing relational definitions
*** TODO Include examples of finite and aleph₀ joins
*** TODO Provide link to E. F. Codd’s original papers