nLab absorption monoid

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Contents

Context

Algebra

Monoid theory

Contents

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,)(M,1,\cdot) that is also an absorption magma (M,0)(M,0).

Equivalently, it is a monoid object in the category of pointed sets, since left and right multiplication \cdot by any element xx preserves the point 00.

Properties

Initial and terminal absorption monoids

The initial absorption monoid is the boolean domain 𝟚\mathbb{2} with elements 0𝟚0 \in \mathbb{2} representing false, 1𝟚1 \in \mathbb{2} representing true, and ()():𝟚×𝟚𝟚(-)\cdot(-):\mathbb{2} \times \mathbb{2} \to \mathbb{2} representing conjunction.

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

Absorption monoid homomorphisms

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

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

Ideals and anti-ideals

A two-sided ideal of an absorption monoid MM is a subset II of MM such that

  • 0I0 \in I
  • for all elements aMa \in M and bMb \in M, if abIa \cdot b \in I, then either aIa \in I or bIb \in I.

A two-sided anti-ideal of an absorption monoid MM is a subset AA of MM such that

  • 0I0 \notin I
  • for all elements aMa \in M and bMb \in M, if aIa \in I and bIb \in I, then abIa \cdot b \in I.

Quotient absorption monoids

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

Invertible elements

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

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

Division monoids

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

Regular elements

An element aMa \in M is a regular element, cancellative element, or cancellable element if for all elements bMb \in M and cMc \in M, b=cb = c if and only if ab=aca \cdot b = a \cdot c and ca=cbc \cdot a = c \cdot b.

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

Integral monoids

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

Ore sets and Ore absorption monoids

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

  • for all aSa \in S and bMb \in M, there exists cSc \in S and dMd \in M such that ad=bca \cdot d = b \cdot c
  • for all aSa \in S and bMb \in M, there exists cSc \in S and dMd \in M such that da=cbd \cdot a = c \cdot b

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

Localization and group completion

The localization of an Ore integral monoid MM at Reg(M)\mathrm{Reg}(M) is a division monoid. The localization of an absorption monoid at 00 is the trivial group; thus, the group completion of any absorption monoid is the trivial group.

Actions and modules

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

  • for all elements pPp \in P, α L(1,p)=p\alpha_L(1, p) = p
  • for all elements aMa \in M, bMb \in M, and cPc \in P, α L(a,α L(b,c))=α L(ab,c)\alpha_L(a, \alpha_L(b, c)) = \alpha_L(a \cdot b, c)
  • for all elements pPp \in P, α L(0,p)=0\alpha_L(0, p) = 0
  • for all elements aMa \in M, α L(a,0)=0\alpha_L(a, 0) = 0

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

  • for all elements pPp \in P, α R(p,1)=p\alpha_R(p, 1) = p
  • for all elements aMa \in M, bMb \in M, and cPc \in P, α R(α R(c,a),b)=α R(c,ab)\alpha_R(\alpha_R(c, a), b) = \alpha_R(c, a \cdot b)
  • for all elements pPp \in P, α R(p,0)=0\alpha_R(p, 0) = 0
  • for all elements aMa \in M, α R(0,a)=0\alpha_R(0, a) = 0

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

  • for all elements pPp \in P, α(1,p,1)=p\alpha(1, p, 1) = p
  • for all elements aMa \in M, bMb \in M, cPc \in P, dNd \in N, eNe \in N, α(a,α(b,c,d),e)=α L(ab,c,de)\alpha(a, \alpha(b, c, d), e) = \alpha_L(a \cdot b, c, d \cdot e)
  • for all elements pPp \in P and dNd \in N, α L(0,p,d)=0\alpha_L(0, p, d) = 0
  • for all elements aMa \in M and dNd \in N, α L(a,0,d)=0\alpha_L(a, 0, d) = 0
  • for all elements aMa \in M and pPp \in P, α L(a,p,0)=0\alpha_L(a, p, 0) = 0

Pointed sets equipped with left or right MM-actions are called left or right MM-modules, and pointed sets equipped with MM-NN-biactions are called MM-NN-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 prime: ×\mathrm{prime}:\mathbb{N} \to \mathbb{N}^\times. ×\mathbb{N}^\times has decidable equality. The localization of ×\mathbb{N}^\times at the image of prime\mathrm{prime}, or equivalently at the non-zero elements of ×\mathbb{N}^\times, is the multiplicative monoid of the non-negative rational numbers, 0 ×\mathbb{Q}_{\geq 0}^\times.

Other examples

algebraic structureoidification
magmamagmoid
pointed magma with an endofunctionsetoid/Bishop set
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
anti-involutive monoiddagger category
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
cancellative monoidcancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
monoidal poset2-poset
strict monoidal groupoid?strict (2,1)-category
strict 2-groupstrict 2-groupoid
strict monoidal categorystrict 2-category
monoidal groupoid(2,1)-category
2-group2-groupoid/bigroupoid
monoidal category2-category/bicategory

Last revised on March 17, 2023 at 15:01:21. See the history of this page for a list of all contributions to it.