adj

• (mathematics) Of an algebraic structure (usually a group or algebra), having a commutative defining operation.
  "The interesting aspect here is that U₃ is irreducible, even though all irreps over the complexes are one-dimensional because ℤ₄ is abelian."
• (mathematics) Of a binary function, commutative.

adj

• (mathematics, group theory, dated) Alternative letter-case form of abelian.
  "Ex. 2. Show by §§ 187, 188 that there can be no transitive Abelian group of prime degree other than the cyclic group, and that there is no irreducible Abelian equation of prime degree other than the cyclic equation."

noun

• (Christianity, historical) A member of a sect in fourth-century Africa mentioned by St. Augustine, who states that they married but lived in continence after the manner, as they claimed, of Abel.
  "1826, unnamed translator, Pierre Bayle, An Historical and Critical Dictionary [1697, Dictionnaire Historique et Critique; 1709, Historical and Critical Dictionary], Selected and Abridged, Volume 1, page 50, The Abelians were only a moderate sort of Encratites and Novatians, who absolutely condemned matrimony, while the Abelians approved of and retained it."