Relational Operations in FOLE

TL;DR

This paper demonstrates how FOLE can represent relational operations from database theory, providing a clear, implementable framework based on basic components and their compositions.

Contribution

It introduces a principled, component-based approach to express and classify relational operations within FOLE, bridging logical representation and database operations.

Findings

Identifies 9 basic components for relational operations in FOLE.

Classifies composite operations into limit-like, colimit-like, and unorthodox categories.

Provides a flowchart-based illustration of composite operations.

Abstract

This paper discusses relational operations in the first-order logical environment {FOLE}. Here we demonstrate how FOLE expresses the relational operations of database theory in a clear and implementable representation. An analysis of the representation of database tables/relations in FOLE reveals a principled way to express the relational operations. This representation is expressed in terms of a distinction between basic components versus composite relational operations. The 9 basic components fall into three categories: reflection (2), Booleans or basic operations (3), and adjoint flow (4). Adjoint flow is given for signatures (2) and for type domains (2), which are then combined into full adjoint flow. The basic components are used to express various composite operations, where we illustrate each of these with a flowchart. Implementation of the composite operations is then expressed in an input/output table containing four parts: constraint, construction, input, and output. We explain how limits and colimits are constructed from diagrams of tables, and then classify composite relational operations into three categories: limit-like, colimit-like and unorthodox.