Help:Multiple Definitions

Different sources may give different Definitions for the same thing. Because aims to be accessible to a large public, equivalent definitions are treated equally.

Introduction
Instead of choosing only one definition as the "main" definition and taking the other definitions as theorems, on equivalent definitions are bundled on the same definition page. Their equivalence is proved on a separate page. This has multiple advantages:


 * Every student feels at home at, regardless of the definition their textbook uses.
 * Seeing different definitions in one glance, increases the chance for a visitor to understand the concept being defined.
 * Different viewpoints on the same concept makes more coherent, without making the pages appear too encyclopaedic.
 * The easy-to-verify versus usefulness trade-off: The ideal definition is easy to verify in practice, yet strong enough to deduce many interesting properties. These two are often incompatible, and multiple definitions allow to profit from both advantages.

Page Structure
The structure of page with multiple equivalent definitions is roughly as follows:

Definition
Let ... ... Let ...

Second transcluded definition
...

Also see

 * Equivalence of Definitions of Concept that is Defined

As with multiple proofs, the introductory part is copied identically on the transcluded subpages, but not inside the  tags.

A link to a proof showing their equivalence (see below) is placed in the Also see section, preferably as the first item in the list, so as to make it easier to find the link.

Pages that severly deviate from this structure have to be refactored accordingly.

Equivalence Proofs
The equivalence of the definitions has to be proved on a separate page. More instructions at Help:Definition Equivalences.

Equivalent Characterizations
One should not exaggerate with multiple definitions. Especially if there are many equivalent characterizations which are otherwise not likely to contribute to the comprehensibility of the concept, it is better to set up a page that bundles all equivalent characterizations. See Help:Equivalence Proofs. You may also want to enhance the corresponding Also see section or Also defined as section.

As a general rule, when a definition can be found in a textbook, it is considered appropriate to add it as an alternative definition rather than a theorem.

If the equivalence proof requires some theory to be built on one of the definitions, it is often a better idea to treat them as characterizations, not equivalent definitions. Consider that, strictly speaking, one cannot prove theorems about something before the definition is settled. But there is no general rule, and such difficult cases have to be considered case by case.

Related definitions
While unrelated definitions with the same name have to be disambiguated, very closely related definitions bearing a same name can be treated in the same way as generalizations.

Generalizations
It is not uncommon that a definition has one or more generalizations. Naturally, this does not mean that only the most general definition should be given:


 * aims to be accessible to people with different background
 * Nobody knows if an even more general definition will be invented some day.
 * The way in which definitions generalize may be not totally ordered.

If one definition applies to a significantly more general setting, it should not be treated as just another equivalent definition. Instead, the less and more general definitions are placed on separate pages. Depending on the situation, there are roughly two ways to do this:


 * 1) Transclude the pages on a common page. They can either be transcluded as subpages, which is appropriate in most cases, or as independent pages.
 * 2) Set up a disambiguation page. This becomes especially more convenient if there is a large amount of related definitions, which would otherwise compromise the conciseness of the page on which everything is transcluded.

Nonequivalent definitions

 * See also Help:Also defined as

Also see

 * Help:Multiple Proofs
 * Help:Name Conflict