Wikidata:Property proposal/algebraic properties

identity element

Originally proposed at Wikidata:Property proposal/Natural science

   Done: identity element (P8864) (Talk and documentation)

Description	value of the identity element of the mathematical operation
Represents	identity element
Data type	Item
Domain	binary operation (Q164307)
Allowed values	mathematical object (Q246672)
Example 1	addition (Q32043) → zero (Q204)
Example 2	multiplication (Q40276) → 1 (Q199)
Example 3	function composition (Q244761) → identity (Q254474)
Example 4	matrix addition (Q2264115) → zero matrix (Q338028)
Example 5	matrix multiplication (Q1049914) → identity matrix (Q193794)

More examples: [1].

mathematical inverse

Originally proposed at Wikidata:Property proposal/Natural science

   Done: mathematical inverse (P8865) (Talk and documentation)

Description	the inverse element with respect to binary operation given as a qualifier
Represents	mathematical object (Q246672)
Data type	Item
Domain	mathematical object (Q246672)
Allowed values	mathematical object (Q246672)
Example 1	2 (Q200) mathematical inverse ½ (Q2114394) / relative to (P2210) multiplication (Q40276)
Example 2	−1 (Q310395) mathematical inverse 1 (Q199) / relative to (P2210) addition (Q32043)
Example 3	Laplace transform (Q199691) mathematical inverse inverse Laplace transform (Q2162701) / relative to (P2210) function composition (Q244761)
Example 4	sine (Q152415) mathematical inverse arcsine (Q674517) / relative to (P2210) function composition (Q244761)
Example 5	addition (Q32043) mathematical inverse subtraction (Q40754)

Other examples

• inverse Legendre transform (Q98967608) <=> Legendre transform (Q30587593)
• convolution (Q210857) <=> deconvolution (Q1183700)
• Fourier transform (Q6520159) <=> inverse Fourier transform (Q98968150)

has operator

Originally proposed at Wikidata:Property proposal/Natural science

   Done: has operator (P8866) (Talk and documentation)

Description	mathematical operator associated with this algebraic structure
Data type	Item
Domain	algebraic structure (Q205464)
Allowed values	operator (Q131030)
Example 1	additive group (Q4681347) → addition (Q32043)
Example 2	symmetric group (Q849512) → function composition (Q244761)
Example 3	matrix ring (Q2915729) → matrix addition (Q2264115), matrix multiplication (Q1049914)
Example 4	vector field (Q186247) → vector addition (Q55091432), scalar multiplication (Q126736)

Motivation

Algebriac groups 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 set (Q36161) and an invertible binary operation (Q164307), which has a unique identity element (Q185813). 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 has part (P527), but it results in set of real numbers (Q26851380) has part (P527) zero (Q204) / subject has role (P2868) identity element (Q185813) / relative to (P2210) addition (Q32043) and set of real numbers (Q26851380) has part (P527) addition (Q32043) / subject has role (P2868) group operation (Q99839939) but that is super convoluted and querying it would be difficult. Similarly, we can model "mathematical inverse" as −1 (Q310395) opposite of 1 (Q199) / relative to (P2210) addition (Q32043) / criterion used (P1013) mathematical inverse 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. set of real numbers identity element zero / with respect to addition
2. set of real numbers has operation addition / identity element zero
3. set of real numbers has operation addition addition identity element zero

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 $I\not \in S$. I don't know if this is possible or not.)

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

Discussion

<ping project should not be used in an indented reply> Opensofias
Tobias1984
Arthur Rubin
Cuvwb
TomT0m
Physikerwelt
Lymantria
Bigbossfarin
Infovarius
Helder
PhilMINT
Malore
Nomen ad hoc
Lore.mazza51
Wikisaurus
The Anome
The-erinaceous-one
Daniel Mietchen
Haansn08
Xenmorpha Notified participants of WikiProject Mathematics

 Support 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)

@ArthurPSmith: 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 Laplace transform (Q199691) mathematical inverse inverse Laplace transform (Q2162701) / relative to (P2210) function composition (Q244761), 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 Laplace transform (Q199691) mathematical inverse inverse Laplace transform (Q2162701) / relative to (P2210) function composition (Q244761) is correct, so now all the "mathematical inverse" examples have qualifiers. — The Erinaceous One 🦔 21:45, 3 October 2020 (UTC)

@Jura1, Ederporto: would one of you be able to create these properties? — The Erinaceous One 🦔 09:06, 11 November 2020 (UTC)

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

@The-erinaceous-one, ArthurPSmith, Tinker Bell:  Done identity element (P8864), mathematical inverse (P8865), has operator (P8866) Pamputt (talk) 17:07, 26 November 2020 (UTC)