# Difference between revisions of "Subwiki:Relations in definition article"

Definition articles, or article defining new terms, have relation sections. These relation sections describe how the term being defined (henceforth, called the definiendum) is related to other terms and concepts.

## Relation with similarly typed terms

The type of a term is the specific role that term plays. For instance, some terms may be of the property type: prime number is a property of natural numbers. Some terms may be of the species type, so mangifera indica is a species of plant. (Learn more about typing at Subwiki:Type-based organization).

The relation with similarly typed terms describes how the definiendum is related to terms of the same type.

### Stronger and weaker

For properties over some context space, one can talk of stronger properties and weaker properties, to describe the relation with other properties over the same context space. Thus, for instance, the property of natural numbers of being a perfect fourth power is stronger than the property of being a perfect square, while the property of being a perfect power is weaker than the property of being a perfect square.

Something similar is true for extra structures placed. A stronger structure in this case is a structure that carries more information (or more constraints) than a weaker structure, and from which the weaker structure can be recovered. For instance, the structure of a Riemannian manifold is stronger than the structure of a topological manifold, and the structure of a group is stronger than the structure of a magma (a set with a binary operation).

We typically say that the stronger property (resp., structure) implies (resp., gives rise to) the weaker property (resp., structure). Here's how this is presented in the case of properties:

• In a section titled Relation with other properties, there are subsections titled Stronger properties and Weaker properties.
• In the section titled Stronger properties, there are bullet points listing the stronger properties. When available or desired, each property is accompanied by a short description either of the property or of why it is stronger; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties.
• In the section titled Weaker properties, there are bullet points listing the weaker properties. When available or desired, each property is accompanied by a short description either of the property or of why it is weaker; there is usually also a link to a more complete proof, both of the implication, and of why the converse implication fails in general. There may also be a link to a survey article comparing the properties.
• There may also be a section titled Related properties or Incomparable properties that lists properties that are neither stronger nor weaker but are still closely related.

A similar format is followed for stronger/weaker structure and for stronger notions and weaker notions in other senses.

### Conjunctions and disjunctions

Sometimes new terms are obtained by taking the definiendum and ANDing it with something else. For instance, we can take the AND of the property of being a prime number and the property of being an odd number, and get the property of being an odd prime. We can take the OR of being a number that is 1 mod 3 and a number tha tis 2 mod 3, to get the property of being a number that is not a multiple of 3.

For property articles, conjunctions and disjunctions are typically handled in separate subsections under the Relation with other properties section. The section with conjunctions is titled Conjunction with other properties while the section on disjunctions is titled Disjunction with other properties.

## Relation with terms of related but different types

Apart from being related with terms of the same type, the definiendum may also be related with terms of different types. These relations may be somewhat more complicated.

### A broad banner of related properties

Suppose a subgroup property has some related group properties. Then, these group properties are listed in a subsection titled Related group properties under the section Relation with other properties.