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unital magma
Redirected from "unital algebra object".
Contents
Context
Algebra
- algebra, higher algebra
- universal algebra
- monoid, semigroup, quasigroup
- nonassociative algebra
- associative unital algebra
- commutative algebra
- Lie algebra, Jordan algebra
- Leibniz algebra, pre-Lie algebra
- Poisson algebra, Frobenius algebra
- lattice, frame, quantale
- Boolean ring, Heyting algebra
- commutator, center
- monad, comonad
- distributive law
Group theory
Ring theory
Module theory
Contents
Definition
A magma is called unital if it has an identity element , hence an element such that for all it satisfies the equation
holds. The identity element is idempotent.
Some authors take a magma to be unital by default (cf. Borceux-Bourn Def. 1.2.1).
There is also a possibly empty version, where the identity element is replaced with a constant function such that for all , and .
Properties
The Eckmann-Hilton argument holds for unital magmas: two compatible ones on a set must be equal, associative and commutative.
Examples
Examples include unital rings etc.
Generalizations
This concept could be generalized from the category of sets to any monoidal category:
A unital magma object or unital algebra object in a monoidal category is an object with morphisms and such that the following diagrams commute:
where and are the left and right unitors of the monoidal category.
In the category of modules, unital magma objects are called nonassociative unital algebras, and in the category of abelian groups, unital magma objects are called nonassociative rings.
Last revised on August 21, 2024 at 02:21:32.
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