nLab absorption monoid

Definition

An absorption monoid or annihilation monoid is a monoid $(M,1,\cdot)$ that is also an absorption magma $(M,0)$.

Properties

Zero divisors

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

Localization and group completion

The localization of an absorption monoid at $0$ is the trivial group. Because of this, the group completion of any absorption monoid is the trivial group. This is why one speaks of division monoids instead of groups in the context of absorption monoids, and in particular, why the additive identity element in any nontrivial field has no multiplicative inverse.

Examples

• The extended natural numbers $(\bar{\mathbb{N}}, 0, +, \infty)$ are an absorption monoid.

• Every integral monoid is an absorption monoid.

• The multiplicative monoid of every rig is an absorption monoid.

• Every join-semilattice with a top element and evety meet-semilattice with a bottom element is an absorption monoid.

Related concepts

• monoid

• integral monoid

• zero divisor

• rig, ring, lattice, field

algebraic structure	oidification
truth value	preorder
magma	magmoid
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory