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Submission declined on 9 June 2026 by GearsDatapacks (talk). Still needs more work on the lead section. The first sentence should explain what a prime relation is, rather than how they arise or how they are used. If you would like to continue working on the submission, click on the "Edit" tab at the top of the window. If you have not resolved the issues listed above, your draft will be declined again and potentially deleted. If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors. Please do not remove reviewer comments or this notice until the submission is accepted. Where to get help If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews. If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed. How to improve a draft Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia. Help:Wikitext – how to use the markup Help:Referencing for beginners – how to include references Wikipedia:Article development – how to develop your article Wikipedia:Writing better articles – how to improve your article Wikipedia:Verifiability – make sure your article includes reliable third-party sources You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Add tags to your draft Editor resources Easy tools: Citation bot (help) | Advanced: Fix bare URLs Declined by GearsDatapacks 6 hours ago. Last edited by Leorino 6 hours ago. Reviewer: Inform author. This draft has been resubmitted and is currently awaiting re-review.

Submission declined on 6 June 2026 by Stuartyeates (talk). Needs a better lede. Imagine you're explaining this to a teenager you're seated next to on a long train journey. Also explain how it differs from Draft:Prime Strongly Regular Relation. Declined by Stuartyeates 2 days ago.

• Comment: In accordance with Wikipedia's Conflict of interest guideline, I disclose that I have a conflict of interest regarding the subject of this article. Leorino (talk) 17:28, 6 June 2026 (UTC)

The Prime relation is a special type of equivalence relation on hyperrings (and hypermodules), which can be regarded as a congruence relation modulo a prime hyperideal (or a prime sub-hypermodule). Prime relations arise in pure mathematics, particularly in algebra, algebraic geometry, algebraic topology, and the theory of algebraic hyperstructures. Through prime relations, hyperrings can be reduced to integral domains (rings without zero divisors), while geometric and topological structures arising from hyperrings are translated into their corresponding classical geometric and topological structures. Every prime relation on a hyperring is in one-to-one correspondence with a prime hyperideal. More precisely, if ${\textstyle \rho }$ is a prime relation on a hyperring ${\textstyle R}$, then a prime hyperideal can be derived from ${\textstyle \rho }$. Conversely, if ${\textstyle P}$ is a prime hyperideal of ${\textstyle R}$, then the congruence relation modulo ${\textstyle P}$ is a prime relation on ${\textstyle R}$. As a consequence, the Zariski topology determined by the prime hyperideals of a hyperring ${\textstyle R}$ is homeomorphic to the topology induced by the prime relations on ${\textstyle R}$.

A krasner hyperring is an algebraic structure ${\textstyle (R,+,\cdot )}$ which satisfies following axioms:

• ${\textstyle (R,+)}$ is a canonical hypergroup, i.e.,

1. for every ${\textstyle x,y,z\in R}$, ${\textstyle x+(y+z)=(x+y)+z}$,
2. for every ${\textstyle x,y\in R}$, ${\textstyle x+y=y+x}$,
3. there exists ${\textstyle 0\in R}$ such that ${\textstyle 0+x=\{x\}}$ for every ${\textstyle x\in R}$,
4. for every ${\textstyle x\in R}$, there exists a unique element ${\textstyle x'\in R}$ such that ${\textstyle 0\in x+x'}$, (${\textstyle x'}$ is represented by ${\textstyle -x}$),

• ${\textstyle z\in x+y}$ implies ${\textstyle y\in -x+z}$ and ${\textstyle x\in z-y}$;
• ${\textstyle (R,\cdot )}$ is a semigroup having zero as a bilaterally absorbing element, i.e., ${\textstyle x\cdot 0=0\cdot x=0}$,
• the multiplication is distributive with respect to the hyperoperation ${\textstyle +}$.^[1]

A Krasner hyperring ${\textstyle (R,+,\cdot )}$ is called commutative (with unite element) if ${\textstyle (R,\cdot )}$ is a commutative semigroup (with unite element).^[1]

An equivalence relation ${\textstyle \rho }$ on a Krasner hyperring ${\textstyle R}$ is called strongly regular relation if for every ${\textstyle (x,y)\in \rho }$ and ${\textstyle z\in R}$ we have:

• ${\textstyle (x+z)\ {\bar {\bar {\rho }}}\ (y+z)}$, (i.e. ${\textstyle (s,t)\in \rho }$, for every ${\textstyle s\in x+z}$ and ${\textstyle t\in y+z}$),
• ${\textstyle xz\ \rho \ yz}$, (i.e. ${\textstyle (xz,yz)\in \rho }$).^[2]

Definition^[3]: A strongly regular relation ${\textstyle \rho }$ on a Krasner hyperring ${\textstyle R}$ is called prime, if for every ${\textstyle x}$ and ${\textstyle y}$ in ${\textstyle R}$, we have: $${\displaystyle (xy,0)\in \rho \ \Rightarrow \ (x,0)\in \rho \ \ or\ \ (y,0)\in \rho .}$$

A subhyperring ${\textstyle I}$ of a Krasner hyperring ${\textstyle R}$ is a left (right) hyperideal of ${\textstyle R}$ if ${\textstyle r\cdot a\in I}$ (${\textstyle a\cdot r\in I}$), for all ${\textstyle r\in R,a\in I}$. In particular, I is called a hyperideal if ${\textstyle I}$ is both a left and a right hyperideal.^[2]

A proper hyperideal ${\textstyle P}$ of a Krasner hyperring ${\textstyle R}$ is called prime if for every pair of hyperideals ${\textstyle A}$ and ${\textstyle B}$ of ${\textstyle R}$: $${\displaystyle AB\subseteq P\ \Rightarrow \ A\subseteq P\ \ or\ \ B\subseteq P}$$.

Lemma^[3]: If ${\textstyle \rho }$ is a strongly regular relation on Krasner hyperring ${\textstyle R}$, then ${\textstyle \rho }$ is prime if and only if $${\displaystyle \rho (0_{R})=\{x\in R\ |\ (x,0_{R})\in \rho \}}$$ is prime hyperideal of ${\textstyle R}$.

Example^[3]:Consider the Krasner hyperring ${\textstyle R}$ as follows:

Let ${\textstyle X=\{0,c,a,d\}}$ and ${\textstyle Y=\{1,f,b,e\}}$, then by definition, $${\displaystyle \rho _{1}=P(X)\cup P(Y)}$$ is a prime relation and it can also be seen that ${\textstyle \rho _{1}(0)}$ is a prime hyperideal of ${\textstyle R}$. On the other hand, $${\displaystyle \rho _{2}=\{(a,d),(d,a),(b,e),(e,b),(1,f),(f,1),(0,c),(c,0)\}\cup \Delta _{R}}$$ is not prime, because ${\textstyle (ad,0)\in \rho }$ but ${\textstyle (a,0),(d,0)\notin \rho }$. Furthermore, ${\textstyle \rho _{2}(0)}$ is not a prime hyperideal of ${\textstyle R}$ either.

The prime relations were first introduced in 2025 by B. Afshar, while working on Zariski topology of Krasner hyperrings, to express algebraic varieties in terms of strongly regular relations.^[3] These relations help to more easily examine the connection of hyperstructural algebraic geometry with the classical state, and in addition, they have many applications in the study of sheaves of algebraic hyperstructures.