nLab absorption monoid

Idea

Absorption monoids are the monoid objects in pointed sets, in the same way that rings are the monoid objects in abelian groups. Thus, the theory of absorption monoids and the theory of rings are very similar to each other, except that rings have additive structure whereas absorption monoids do not have additive structure.

Definition

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

Equivalently, it is a monoid object in the category of pointed sets, since left and right multiplication $\cdot$ by any element $x$ preserves the point $0$.

Properties

Initial and terminal absorption monoids

The initial absorption monoid is the boolean domain $\mathbb{2}$ with elements $0 \in \mathbb{2}$ representing false, $1 \in \mathbb{2}$ representing true, and $(-)\cdot(-):\mathbb{2} \times \mathbb{2} \to \mathbb{2}$ representing conjunction.

The terminal absorption monoid is the trivial monoid $\mathbb{1}$, the monoid whose underlying set is a singleton. The trivial monoid is also strictly terminal.

Absorption monoid homomorphisms

Given absorption monoids $M$ and $N$, an absorption monoid homomorphism is a function $h:M \to N$ such that

• $h(0) = 0$
• $h(1) = 1$
• for all $a \in M$ and $b \in M$, $h(a \cdot b) = h(a) \cdot h(b)$.

Ideals and anti-ideals

A two-sided ideal of an absorption monoid $M$ is a subset $I$ of $M$ such that

• $0 \in I$
• for all elements $a \in M$ and $b \in M$, if $a \cdot b \in I$, then either $a \in I$ or $b \in I$.

A two-sided anti-ideal of an absorption monoid $M$ is a subset $A$ of $M$ such that

• $0 \notin I$
• for all elements $a \in M$ and $b \in M$, if $a \in I$ and $b \in I$, then $a \cdot b \in I$.

Quotient absorption monoids

Given an absorption monoid $M$ and a two-sided ideal $I$, the quotient of $M$ by $I$ is the initial absorption monoid $M/I$ with absorption monoid homomorphism $i:M \to M/I$ such that for all elements $a \in I$, $i(a) = 0$.

Invertible elements

An element $a \in M$ is an invertible element or a unit if there exists an element $b \in M$ such that $a \cdot b = 1$ and $b \cdot a = 1$.

The set of invertible elements $M^\times$ in an absorption monoid $M$ is always closed under multiplication; i.e. $M^\times$ is a submonoid of $M$. In fact, since every element is invertible, $M^\times$ forms a subgroup of $M$, called the group of units.

Division monoids

An absorption monoid $M$ is a division monoid if every non-invertible element in $M$ is equal to zero. $M$ is Heyting if there is a tight apartness relation on $M$ such that every invertible element is apart from zero, and $M$ is discrete if every element in $M$ is either zero or invertible.

Regular elements

An element $a \in M$ is a regular element, cancellative element, or cancellable element if for all elements $b \in M$ and $c \in M$, $b = c$ if and only if $a \cdot b = a \cdot c$ and $c \cdot a = c \cdot b$.

The set of regular elements $\mathrm{Reg}(M)$ in an absorption monoid $M$ is always closed under multiplication; i.e. $\mathrm{Reg}(M)$ is a submonoid of $M$.

Integral monoids

An absorption monoid $M$ is an integral monoid if every non-regular element in $M$ is equal to zero. $M$ is Heyting if there is a tight apartness relation on $M$ such that every regular element is apart from zero, and $M$ is discrete if every element in $M$ is either zero or regular.

Ore sets and Ore absorption monoids

Given an absorption monoid $M$, an Ore set is a submonoid $S$ of $\mathrm{Reg}(M)$ such that every element of $S$ satisfies the left and right Ore conditions:

• for all $a \in S$ and $b \in M$, there exists $c \in S$ and $d \in M$ such that $a \cdot d = b \cdot c$
• for all $a \in S$ and $b \in M$, there exists $c \in S$ and $d \in M$ such that $d \cdot a = c \cdot b$

A absorption monoid is an Ore absorption monoid if $\mathrm{Reg}(M)$ is an Ore set.

Localization and group completion

The localization of an Ore integral monoid $M$ at $\mathrm{Reg}(M)$ is a division monoid. The localization of an absorption monoid at $0$ is the trivial group; thus, the group completion of any absorption monoid is the trivial group.

Actions and modules

Given an absorption monoid $M$, an left $M$-action on a pointed set $(P, 0)$ is an ternary function $\alpha_L:M \times P \to P$ such that:

• for all elements $p \in P$, $\alpha_L(1, p) = p$
• for all elements $a \in M$, $b \in M$, and $c \in P$, $\alpha_L(a, \alpha_L(b, c)) = \alpha_L(a \cdot b, c)$
• for all elements $p \in P$, $\alpha_L(0, p) = 0$
• for all elements $a \in M$, $\alpha_L(a, 0) = 0$

A right $M$-action on a pointed set $(P, 0)$ is a binary function $\alpha_R:P \times M \to P$ such that:

• for all elements $p \in P$, $\alpha_R(p, 1) = p$
• for all elements $a \in M$, $b \in M$, and $c \in P$, $\alpha_R(\alpha_R(c, a), b) = \alpha_R(c, a \cdot b)$
• for all elements $p \in P$, $\alpha_R(p, 0) = 0$
• for all elements $a \in M$, $\alpha_R(0, a) = 0$

Given absorption monoids $M$ and $N$, an $M$-$N$-biaction on a pointed set $(P, 0)$ is a ternary function $\alpha:M \times P \times N \to P$ such that:

• for all elements $p \in P$, $\alpha(1, p, 1) = p$
• for all elements $a \in M$, $b \in M$, $c \in P$, $d \in N$, $e \in N$, $\alpha(a, \alpha(b, c, d), e) = \alpha_L(a \cdot b, c, d \cdot e)$
• for all elements $p \in P$ and $d \in N$, $\alpha_L(0, p, d) = 0$
• for all elements $a \in M$ and $d \in N$, $\alpha_L(a, 0, d) = 0$
• for all elements $a \in M$ and $p \in P$, $\alpha_L(a, p, 0) = 0$

Pointed sets equipped with left or right $M$-actions are called left or right $M$-modules, and pointed sets equipped with $M$-$N$-biactions are called $M$-$N$-bimodules.

Examples

The multiplicative monoid of the natural numbers

The multiplicative monoid of the natural numbers $\mathbb{N}^\times$ is the free commutative absorption monoid on the natural numbers, the initial commutative absorption monoid $\mathbb{N}^\times$ with a function $\mathrm{prime}:\mathbb{N} \to \mathbb{N}^\times$. $\mathbb{N}^\times$ has decidable equality. The localization of $\mathbb{N}^\times$ at the image of $\mathrm{prime}$, or equivalently at the non-zero elements of $\mathbb{N}^\times$, is the multiplicative monoid of the non-negative rational numbers, $\mathbb{Q}_{\geq 0}^\times$.

Other 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
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
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