Testwiki:Property proposal/algebraic properties

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More examples: [1].

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Other examples

• Template:Q <=> Template:Q
• Template:Q <=> Template:Q
• Template:Q <=> Template:Q

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Motivation

Algebriac Template:Q-s are an important class of mathematical object. In Wikidata, there are 184 instances [2] and 585 subclasses [3] of algebraic groups, but they are lacking the most basic information to describe them. A group consists of a mathematical Template:Q and an invertible Template:Q, which has a unique Template:Q. In order to model this information, I propose creating three new properties: identity element, mathematical inverse, and has operator. (These properties are applicable beyond just groups, however. Multiple examples are listed above.)

Existing properties are inadequate or cumbersome for modeling these relationships. We could model "identity element" and "has operator" with existing properties by using Template:P, but it results in Template:Statement and Template:Statement but that is super convoluted and querying it would be difficult. Similarly, we can model "mathematical inverse" as Template:Statement but, again, this is awkward and also prevents us from enforcing expected relationships, such as the symmetry of inverses.

I considered three ways of modeling identity element:

1. Template:Statement
2. Template:Statement
3. Template:Statement Template:Statement

Option 1 was dismissed because an "identity" is an identity of an of the operator rather than the set, so the modeling should reflec that. Option 2 is an improvement over option 1---and we might want to use "identity element" as a qualifier sometimes (please discuss)---but I think option 3 is best choice because it is simple and will prevent duplication if the same operator is used on multiple structures. (Option 3 might be problematic, however, if some algebraic structure, let's call it $S$, uses an operator that has an identity element $I$, but $I\in ̸S$. I don't know if this is possible or not.)

— The Erinaceous One 🦔 10:53, 2 October 2020 (UTC)

Discussion

Template:S for all of these; however I have a question about the "mathematical inverse" proposal - two of your examples have the qualifier and two don't - does that suggest these are two distinct properties, or is there a better way to describe this? ArthurPSmith (talk) 20:51, 2 October 2020 (UTC)

Template:Ping No, it should only be one property. I've added a qualifier for the sine/arccosine example and qualfier would work on the Laplace transform example, but I'm not sure what the right qualifier value is. We could use Template:Statement, but I'm not sure if it's accuracte to talk about the function composition of integral transforms. (I.e. is an integral transformation a function on functions?) — The Erinaceous One 🦔 22:14, 2 October 2020 (UTC)

I've thought about it more and Template:Statement is correct, so now all the "mathematical inverse" examples have qualifiers. — The Erinaceous One 🦔 21:45, 3 October 2020 (UTC)

Template:Ping would one of you be able to create these properties? — The Erinaceous One 🦔 09:06, 11 November 2020 (UTC)

• Template:S --Tinker Bell ★ ♥ 02:40, 17 November 2020 (UTC)

Template:Ping Template:Done Template:P, Template:P, Template:P Pamputt (talk) 17:07, 26 November 2020 (UTC)