algebrast

An algebrast is a generalized algebraic structure defined on a single set with an additive group and a finite family of binary operations. It formalizes the idea of having multiple products that act on the same additive base and interact through coherence relations.

Formally, an algebrast is a tuple (A, +, {⋆_i}_{i∈I}) where (A, +) is an abelian group and each ⋆_i: A × A → A is bilinear with respect to +: a⋆_i(b+c) = a⋆_i b + a⋆_i c and (a+b)⋆_i c = a⋆_i c + b⋆_i c for all a, b, c ∈ A and i ∈ I. The operations satisfy a chosen set of coherence identities that relate different ⋆_i.

Special cases and relationships: If I = {1} and ⋆_1 is associative and distributive, (A, +, ⋆_1) is a conventional algebra (a ring if a multiplicative identity exists). More generally, algebrasts generalize rings and multi-product algebras and can encode additional structure such as gradings or symmetries through the index set I.

Examples: A standard ring is the simplest algebrast. In thought experiments, one may consider an additive group A equipped with two bilinear products that satisfy a simple coherence law, yielding a two-product algebrast that specializes to familiar cases when the law is strengthened.

Notes: The term "algebrast" is not common in mainstream texts; this article describes a hypothetical framework used in speculative mathematics.