nLab
integral monoid

Contents

Context

Algebra

Monoid theory

Contents

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 MM is an integral monoid if it is nontrivial and the submonoid M\{0}M \backslash \{0\} is a cancellative monoid (i.e., 101 \neq 0 and left and right multiplication by cc is injective if c0c \neq 0, which may be combined as left and right multiplication by cc is injective if and only if c0c \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 cc is injective iff cc is nonzero” in the above definition by “left and right multiplication by cc is injective xor cc 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 010 \neq 1.

Such an integral monoid MM is ‘discrete’ in that it decomposes as a coproduct M={0}M ×M = \{0\} \sqcup M^\times (where M ×M^\times is the submomoid of MM 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}, Z\mathbf{Z}, Q\mathbf{Q}, Z/n\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 00 is a GCD monoid is called an GCD integral monoid.

  • An commutative integral monoid whose largest submonoid not containing 00 is a unique factorization monoid is called an unique factorization integral monoid.

  • An integral monoid whose largest submonoid not containing 00 is a Bézout monoid is called an Bézout integral monoid.

  • An integral monoid whose largest submonoid not containing 00 is a principal ideal monoid is called an principal ideal integral monoid.

  • An integral monoid whose largest submonoid not containing 00 is a group is called an division monoid.

Last revised on June 18, 2021 at 14:41:59. See the history of this page for a list of all contributions to it.