nLab division rig

Definitions

Classically:

Definition

A division rig is a rig in which every nonzero element has a multiplicative inverse and $0 \neq 1$ (which may be combined as: an element is invertible if and only if it is nonzero).

Constructive notions

Division rigs are (arguably) not a purely algebraic notion in that they don't form an algebraic category (see discussion below). For this reason, it should be unsurprising that in constructive mathematics (including the internal logic of a topos) there are different inequivalent ways to define a division rig. In this case the classical definition is not usually the best one; for instance, the non-negative real numbers do not satisfy it. There are several potential replacements with their own advantages and disadvantages.

Definition

If we replace “an element is invertible iff it is nonzero” in Definition by “an element is invertible xor it equals zero” (which is equivalent in classical logic but stronger in constructive logic), then we obtain the notion of discrete division rig. This condition means that every element is either $0$ or invertible, and it also implies that $0\neq 1$.

Such a division rig $R$ is ‘discrete’ in that it decomposes as a coproduct $R = \{0\} \sqcup R^\times$ (where $R^\times$ is the subset of invertible elements). 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 rig of non-negative real numbers (however these are defined), although it is satisfied by the rig of non-negative rationals.

Definition

If we interpret ‘nonzero’ in Definition as a reference to a tight apartness relation, thus defining the apartness relation $\#$ by $x # y$ iff $x - y$ is invertible, then we obtain the notion of Heyting division rig, where the rig operations become strongly extensional functions. In addition to $0\# 1$, the condition then means that every element apart from $0$ is invertible.

This is how ‘practising’ constructive analysts of the Bishop school usually define the simple word ‘division rig’. An advantage is that the (located Dedekind) non-negative real numbers form a Heyting division rig, although (for example) the (less located) non-negative MacNeille real numbers need not form a Heyting division rig; another disadvantage is that this is not a coherent axiom and so cannot be internalized in as many categories.

Every discrete division rig is also a Heyting division rig. A Heyting division rig is a discrete division rig if and only if equality is decidable; it is in this sense that a discrete division rig is ‘discrete’.

Examples

Example

There are the division rigs of:

• non-negative rational numbers,

• non-negative real numbers,

• all division rings, such as fields

• for a division rig $R$, all $R$-division algebras

Related concepts

• rig, integral cancellable rig?

• division ring

• field