DIFFERENCE ALGEBRA v1.0 EN

DESCRIPTION Document No. 4.2 – Difference Algebra v1.0 EN introduces the first formal algebraic framework of Differentics. The document proposes that Difference, rather than the object itself, should be considered the primary mathematical entity. Building upon the foundations established in Formal Differentics v1.0 and Difference Information Theory v1.0, this work develops a systematic algebra of differences through the introduction of Difference Elements (Differents), Difference Magnitudes, Difference Operations, Difference Fields, Difference Metrics, Difference Conservation Laws, and DIXOR Algebra. The theory defines formal operations for composing, transforming, inverting, and recursively generating differences. It further establishes algebraic structures capable of representing distinguishability, transformation, recursion, and diversity generation across physical, informational, computational, biological, and cognitive systems. A central contribution of the work is the formulation of Difference Algebra as a universal symbolic language in which differences become the fundamental objects of mathematical manipulation. Within this framework, traditional entities are treated as manifestations of underlying difference relations, while algebra becomes the study of operations acting upon differences themselves. The document also formalizes the relationship between Difference Algebra and the ADP-TRIFORMER methodology, providing a mathematical foundation for future developments in Difference Geometry, Difference Topology, Difference Dynamics, Difference Computation, and Difference Physics. This publication forms Part 4.2 of the Formal Differentics Mathematical Corpus and serves as the algebraic foundation of the emerging discipline of Differentics.