Algebrades

Algebrades are a fictional class of algebraic structures used in theoretical discussions to explore how multiple binary operations can coexist on a single underlying set. Unlike standard algebras, which typically feature one multiplication along with addition, algebrades are defined by a finite family of bilinear products together with a grading by an abelian monoid. The general idea is to study how different operations interact and to provide a flexible framework for modeling systems with several kinds of composition.

A typical definition (in literature that uses the term) treats an algebrade A over a field F as a vector space equipped with a finite index set I of binary operations ∘_i: A × A → A and a grading A = ⊕_{g∈G} A_g by an abelian monoid G. Each operation respects the grading: if a ∈ A_g and b ∈ A_h, then a ∘_i b ∈ A_{g+h}. A subset of operations may satisfy associativity; others may be derivations or Lie-like brackets, with axioms ensuring a form of coherence among the operations. In many presentations, a distinguished operation ∘_0 is required to be associative and to distribute over the others in a specified way.

Examples and constructions: In the degenerate case where I has a single element and G is trivial, an algebrade reduces to an ordinary associative algebra. More elaborate constructions can be formed by taking a graded algebra and adding a second bilinear operation defined via a fixed linear map, arranged to preserve grading. The notion is deliberately broad, allowing various axiom systems depending on context.

Applications: The term is largely theoretical and appears in speculative discussions, toy models in category theory, and occasional expository notes on multi-operator algebraic systems. It is not part of standard algebra, and there is no single canonical definition.

See also: Algebra, Graded algebra, Multi-algebra, Operad.