nLab absorption magma

Definition

An absorption magma or annihilation magma is a magma $(M,\cdot)$ with an element $0\in M$ satisfying the absorption/annihilation axioms: for all $a \in M$, $0 \cdot a = 0$ and $a \cdot 0 = 0$.

Equivalently, this is a magma object in the category of pointed sets.

Properties

Zero divisors

A non-zero element $a \in M$ is a zero divisor if thete exists a non-zero element $b \in M$ such that $a \cdot b = 0$ or $b \cdot a = 0$.

Examples

• absorption monoids

• Every magma object in the category of commutative unital magmas is an absorption magma. Particular examples of this include the multiplicative monoid in a rig (monoid objects in CMon) or a ring (monoid objects in Ab.

• The multiplicative magmas of the octonions and the sedenions are absorption magmas.

See also

• magma

• zero divisor

• rig, ring, lattice, integral domain field

References

• Wikipedia, Abosrbing element