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. Note: On some of the subject wikis, we have gradually been moving from bullet points to tables, that allow for a more compact and easy-to-read explanation of the two properties, including: the name and meaning of the other property, proof of the implication, proof of failure of the reverse implication (plus all examples), and intermediate notions.
• In the section titled Weaker properties, there are bullet points listing the weaker properties. The format is similar to that for Stronger 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.

Semantic information for stronger and weaker

If $B$ is stronger than $A$, and is mentioned in the article on $A$, the semantic property Property:Weaker than is used. This stores the information that $A$ is weaker than $B$. Conversely, if $A$ is stronger than $B$, the property Property:Stronger than is used. These can be used in semantic search to locate particular properties or structures that are stronger than or weaker than given ones.

Note that conjunctions are stronger than the components, and disjunctions are weaker than the components, and this information is stored, even though conjunctions and disjunctions are usually listed in separate subsections.

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.

Examples

Groupprops

Here are some examples of pages with relations sections:

• Groupprops:Normal subgroup: Here is its relation with other properties section. This has many subsections, including stronger properties, weaker properties, conjunction with other properties. For some of them, there are links to proofs of the implications and of the reverse implications being false, and for some, there are links to survey articles explaining the differences between the properties.
• Groupprops:Pronormal subgroup: Here is its relation with other properties section. This has a list of stronger properties, a list of weaker properties, and a separate subsection titled Conjunction with other properties.
• Groupprops:Complete group: Here is its relation with other properties section. This has a list of stronger properties and a list of weaker properties. For some of these, there are links to proofs of the implications.
• Groupprops:Monoid: Here is its relation with other structures section. This has a list of stronger structures and a list of weaker structures.
• Groupprops:Conjugate subgroups: Here is its relation with other equivalence relations section. This has a list of stronger equivalence relations and weaker equivalence relations.
• Groupprops:Transitive subgroup property: Here is its relation with other metaproperties section. This has a list of stronger metaproperties and a list of weaker metaproperties.

These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in the Special:Ask page:

• For a list of subgroup properties that are stronger than subnormality and weaker than normality:

[[Stronger than::Subnormal subgroup]][[Weaker than::Normal subgroup]]

• For a list of group properties that are stronger than solvable group and weaker than Abelian group:

[[Stronger than::Solvable group]][[Weaker than::Abelian group]]

• We can restrict attention to pivotal subgroup properties that are stronger than normality:

[[Stronger than::Normal subgroup]][[Category:Pivotal subgroup properties]]

• For a list of the subgroup properties that are formed as AND of the property of normality with something else:

[[Conjunction involving::Normal subgroup]][[Category:Subgroup properties]]

• For a list of the transitive subgroup properties that are stronger than the property of normality:

[[Stronger than::Normal subgroup]][[Category:Transitive subgroup properties]]

• For a list of the subgroup properties that are variations of normality, and are stronger than the property of being a subnormal subgroup:

[[Stronger than::Subnormal subgroup]][[Category:Variations of normality]]

Topospaces

Here are some examples of pages with relations sections:

• Topospaces:Normal space: Here is its relation with other properties section. This has two subsections: stronger properties and weaker properties. For some of these, links to proof of the implication are given.
• Topospaces:Closed subset: Here is its [[Topospaces:Closed subset#Relation with other properties}relation with other properties section]]. This has two subsections: stronger properties and weaker properties.

These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in the Special:Ask page:

• For a list of properties that are stronger than Hausdorffness but weaker than normality:

[[Weaker than::Normal space]][[Stronger than::Hausdorff space]]

• For a list of properties that are stronger than Hausdorffness and are variations of Hausdorffness:

[[Stronger than::Hausdorff space]][[Category:Variations of Hausdorffness]]

• For a list of retract-hereditary properties of topological spaces that are weaker than contractibility:

[[Weaker than::Contractible space]][[Category:Retract-hereditary properties of topological spaces]]

Commalg

Here are some examples of pages with a relations section:

• Commalg:Principal ideal domain: This has a relation with other properties section. The section includes subsections titled stronger properties, weaker properties, as well as a subsection giving pairs of properties whose AND gives the property of being a principal ideal domain.
• Commalg:Noetherian ring: This has a relation with other properties section. This section includes subsections titled stronger properties, weaker properties', as well as conjunction with other properties.
• Commalg:Intersection of maximal ideals: This has a relation with other properties section, listing stronger properties as well as weaker properties.

These relations between properties, structures and equivalence relations of various sorts can be used to locate properties. Here are some examples of semantic queries (that can be executed by typing the query term in the Special:Ask page:

• For a list of properties of commutative unital rings that are stronger than being a Noetherian domain but weaker than being a Euclidean domain, try:

[[Weaker than::Euclidean domain]][[Stronger than::Noetherian domain]]