Subwiki:Relations in definition article - Revision history

Vipul: /* Stronger and weaker */

2009-12-30T12:06:37Z

Stronger and weaker

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	* In a section titled '''Relation with other properties''', there are subsections titled '''Stronger properties''' and '''Weaker 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 '''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. ~~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.		* 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.		* 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.

Vipul: /* Examples */

2009-01-17T20:29:54Z

Examples

← Older revision		Revision as of 20:29, 17 January 2009
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	===Commalg===		===Commalg===

			Here are some examples of pages with a relations section:

			* [[Commalg:Principal ideal domain]]: This has a [[Commalg:Principal ideal domain#Relation with other properties\|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 [[Commalg:Noetherian ring#Relation with other properties\|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 [[Commalg:Intersection of maximal ideals#Relation with other properties\|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 [[Commalg:Special:Ask\|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:

			<pre>[[Weaker than::Euclidean domain]][[Stronger than::Noetherian domain]]</pre>

Vipul: /* Examples */

2009-01-17T17:06:53Z

Examples

← Older revision		Revision as of 17:06, 17 January 2009
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	<pre>[[Stronger than::Normal subgroup]][[Category:Pivotal subgroup properties]]</pre>		<pre>[[Stronger than::Normal subgroup]][[Category:Pivotal subgroup properties]]</pre>

	* ~~To look at all~~ the subgroup properties that are formed as '''AND''' of the property of normality with something else:		* For a list of the subgroup properties that are formed as '''AND''' of the property of normality with something else:

	<pre>[[Conjunction involving::Normal subgroup]][[Category:Subgroup properties]]</pre>		<pre>[[Conjunction involving::Normal subgroup]][[Category:Subgroup properties]]</pre>

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

			<pre>[[Stronger than::Normal subgroup]][[Category:Transitive subgroup properties]]</pre>

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

			<pre>[[Stronger than::Subnormal subgroup]][[Category:Variations of normality]]</pre>

	===Topospaces===		===Topospaces===
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	* [[Topospaces:Normal space]]: Here is its [[Topospaces:Normal space#Relation with other properties\|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:Normal space]]: Here is its [[Topospaces:Normal space#Relation with other properties\|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 [[Groupprops:Special:Ask\|the Special:Ask page]]:		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 [[Groupprops:Special:Ask\|the Special:Ask page]]:
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	<pre>[[Weaker than::Normal space]][[Stronger than::Hausdorff space]]</pre>		<pre>[[Weaker than::Normal space]][[Stronger than::Hausdorff space]]</pre>

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

			<pre>[[Stronger than::Hausdorff space]][[Category:Variations of Hausdorffness]]</pre>

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

			<pre>[[Weaker than::Contractible space]][[Category:Retract-hereditary properties of topological spaces]]</pre>

			===Commalg===

Vipul at 01:22, 17 January 2009

2009-01-17T01:22:23Z

@@ Line 46: / Line 46: @@
 ===Groupprops===
-Here are some examples of pages on properties with a '''Relation with other properties''' section:
+Here are some examples of pages with relations sections:
 * [[Groupprops:Normal subgroup]]: Here is its [[Groupprops:Normal subgroup#Relation with other properties|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.
@@ Line 60: / Line 60: @@
 <pre>[[Stronger than::Subnormal subgroup]][[Weaker than::Normal subgroup]]</pre>

Vipul at 23:07, 16 January 2009

2009-01-16T23:07:38Z

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 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==
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 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'''.

Vipul: New page: Definition articles, or article defining new terms, have ''relation'' sections. These relation sections describe how the term being defined (henceforth, call...

2008-08-25T12:10:32Z

New page: Definition articles, or article defining new terms, have ''relation'' sections. These relation sections describe how the term being defined (henceforth, call...

New page

[[Subwiki: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 '''AND'''ing 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'''.