Generative Algebra

This work proposes an axiomatic research framework for generative algebra. If topology answers the question of where an object exists in which space of distinguishability it is localized – then algebra answers the question of what it does when interacting with other objects. Algebraic structures are interpreted through three ontological levels of a single hierarchy: the object as a stable configuration of the tension functional T , the operation as a realization of the convolution operator C, and symmetry as an invariant of the branching operator B. On this basis generative algebra is defined: semigroups, groups, rings, and fields are treated as stability regimes, homomorphism is redefined as preservation of the τ-class, and a Lie group appears as a continuous family of R-symmetries. A separate section is devoted to the generative algebra of objects - the classification of R-stable configurations by their algebraic structure of symmetries. Definitions, proven consequences of classical theory, and programmatic hypotheses are explicitly distinguished throughout the text. The work is part of the Generative Mathematics series and builds on operational set theory (OST) [1] and the published generative topology and generative geometry [2, 3].