Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. Alternative capitalization of abelian
 n. A member of a sect in fourthcentury Africa mentioned by St. Augustine, who states that they married but lived in continence after the manner, as they claimed, of Abel.
from the GNU version of the Collaborative International Dictionary of English
 n. One of a sect in Africa (4th century), mentioned by St. Augustine, who states that they married, but lived in continence, after the manner, as they pretended, of Abel.
from The Century Dictionary and Cyclopedia
 n. A member of a religious sect which arose in northern Africa in the fourth century.
 Of or pertaining to the Norwegian mathematician Niels Henrik Abel (1802–1829).
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

Modules over more general rings do not necessarily have a basis, and those that do are called Abelian Groups

The nice thing about Abelian grapes is that they taste just the same, no matter what order you eat them in.

Abelian group: group theory is the simplest branch of abstract algebra, and one of the basic divisions is between Abelian groups (named for Abel), where the order of operations doesn't matter (they 'commute'), and nonAbelian groups, where order does matter.

Graydon@178: I think the Abelian grape is my favorite.

PS: Keep forgetting to mention how much I like the Abelian grapes.

The Abelian heretics of Africa abstained from women because Abel died virginal.

In fact, if the only issue in the Clay prize were the UV limit of nonAbelian gauge theory, Tadeusz Balaban would probably already have won the million (see references in above review), using renormalization group methods.

Indeed, from a categorical point of view, a Cartesian product in set theory, a direct product of groups (Abelian or otherwise), a product of topological spaces, and a conjunction of propositions in a deductive system are all instances of a categorical product characterized by a universal property.

For example, given a finite Abelian group, how can it be decomposed into a product of certain of its subgroups?

From then on, a specific category of structures, e.g., a category of sheaves over a topological space X, could be seen as a token of an abstract category of a certain type, e.g., an Abelian category.
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