nLab integral monoid

Idea

One should be able to abstract away the additive structure of an integral domain and just focus on the multiplicative structure, yielding the concept of integral monoid.

Definition

An absorption monoid $M$ is an integral monoid if it is nontrivial and the submonoid $M \backslash \{0\}$ is a cancellative monoid (i.e., $1 \neq 0$ and left and right multiplication by $c$ is injective if $c \neq 0$, which may be combined as left and right multiplication by $c$ is injective if and only if $c \neq 0$).

In constructive mathematics

in constructive mathematics, there are different inequivalent ways to define an integral monoid

Discrete integral monoids

Definition

If we replace “left and right multiplication by $c$ is injective iff $c$ is nonzero” in the above definition by “left and right multiplication by $c$ is injective xor $c$ is zero” (which is equivalent in classical logic but stronger in constructive logic), then we obtain the notion of discrete integral monoid. This condition implies that $0 \neq 1$.

Such an integral monoid $M$ is ‘discrete’ in that it decomposes as a coproduct $M = \{0\} \sqcup M^\times$ (where $M^\times$ is the submomoid of $M$ that is cancellative). An advantage is that this is a coherent theory and hence also a geometric theory. A disadvantage is that this axiom is not satisfied (constructively) by the multiplicative monoid of the real numbers (however these are defined), although it is satisfied by the multiplicative monoid of the integers and the multiplicative monoid of the rationals.

Heyting integral monoids

Definition

If we interpret $\neq$ as a tight apartness relation, such that the absorption monoid becomes strongly extensional, then we obtain the notion of Heyting integral monoid.

This is how ‘practising’ constructive analysts of the Bishop school would define the simple word ‘integral monoid’.

An advantage is that the multiplicative monoid of the (located Dedekind) real numbers form a Heyting integral monoid. A disadvantage is that this is not a coherent axiom and so cannot be internalized in as many categories.

Of course, if the underlying set of the monoid has decidable equality —as is true of the multiplicative monoids of $\mathbb{N}$, $\mathbf{Z}$, $\mathbf{Q}$, $\mathbf{Z}/n$, finite fields, etc— then a Heyting integral monoid is a discrete integral monoid.

Examples

• An integral monoid whose largest submonoid not containing $0$ is a GCD monoid is called an GCD integral monoid.

• An commutative integral monoid whose largest submonoid not containing $0$ is a unique factorization monoid is called an unique factorization integral monoid.

• An integral monoid whose largest submonoid not containing $0$ is a Bézout monoid is called an Bézout integral monoid.

• An integral monoid whose largest submonoid not containing $0$ is a principal ideal monoid is called an principal ideal integral monoid.

• An integral monoid whose largest submonoid not containing $0$ is a group is called an division monoid.

Related concepts

• monoid

• absorption monoid

• cancellative monoid

• integral domain, field