Vipul at 21:32, 5 August 2008

2008-08-05T21:32:17Z

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 Properties are a more precise formulation of the general idea of genus and differentia.
 ==Properties==
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 Metaproperties are properties whose context space is itself a property space. In other words, they are properties evaluated for properties.

Vipul: New page: '''Property-theoretic categorization''' is part of the broader property-theoretic paradigm of organization followed on some subject wikis, particularly those in mathematics. Properties ar...

2008-08-05T20:44:50Z

New page: '''Property-theoretic categorization''' is part of the broader property-theoretic paradigm of organization followed on some subject wikis, particularly those in mathematics. Properties ar...

New page

'''Property-theoretic categorization''' is part of the broader property-theoretic paradigm of organization followed on some subject wikis, particularly those in mathematics.

Properties are a more precise formulation of the general idea of genus and differentia.

==Properties==

A ''property'' over a collection of objects is something which every object either ''has'' (or ''satisfies'') or ''does not have'' (or ''does not satisfy'').

Examples from elementary mathematics:

* Being prime is a ''property'' over the collection of natural numbers. Every natural number either ''has'' the property of being prime, or ''does not have'' the property of being prime.
* Being positive is a ''property'' over the collection of real numbers. Every real number either ''is'' positive or is ''not'' positive.
* Being isosceles is a ''property'' over the collection of triangles. Every triangle either ''is'' isosceles or ''is not'' isosceles.

Examples from higher mathematics:

* Being Abelian is a ''property'' over the collection of groups. Every group either ''is'' Abelian or ''is not'' Abelian.
* Being compact is a ''property'' over the collection of topological spaces. Every topological space either ''is'' compact or ''is not'' compact.

The collection of objects over which the property is being evaluted is here termed the ''context space'' for the property.

===Categorization of properties===

All the properties over the same context space are put in a category, and the category is labeled by the context space. For instance all properties over the context space of groups, are called group properties, and are put in a category titled ''Group properties''. A generic element here is termed a ''group property''.

There are two formats for naming the collection of properties over a given context space:

* context space name, followed by the word ''properties'': For instance, ''group properties'', ''subgroup properties''.
* ''properties of'', followed by context space name: For instance, ''properties of topological spaces''.

There is no distinction between the two ways of naming, and one may be chosen over the other for reasons of greater ease of use.

The categories that list all properties over a given context space are termed ''property categories'', and a list of all such categories is found in the category titled ''Properties'' in the subject wiki. For instance, [[Groupprops:Category:Properties]] lists all the property categories on the Group Properties Wiki.

===Further subcategories of the property category===

The category of ''all'' properties over a given context space may often be overwhelmingly huge and diverse. In some cases, this category is divided into subcategories. However, the following general convention is observed: ''all properties are listed both in the parent category and in the subcategories''.

Subcategories are created in many ways:

* Based on metaproperties: properties the properties may or may not satisfy.
* Based on variation, analogy or opposites relationships with an existing pivotal property: For instance, [[Topospaces:Category:Variations of normality]] lists properties of topological spaces obtained by varying the property of being a normal space.
* Based on meta-criteria like degree of importance, implementation in software packages, and historical development.

==Binary relations==
{{fillin}}

==Metaproperties==

Metaproperties are properties whose context space is itself a property space. In other words, they are properties evaluated for properties.

Subwiki:Property-theoretic categorization - Revision history

Vipul at 21:32, 5 August 2008

Vipul: New page: '''Property-theoretic categorization''' is part of the broader property-theoretic paradigm of organization followed on some subject wikis, particularly those in mathematics. Properties ar...