Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 29th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexcreated [[hyperring]]

   created hyperring

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 1st 2010
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexCorrected the proposition about the structure of $Hom(\mathbb{Z}[X], \mathbf{S})$.

   Corrected the proposition about the structure of $Hom(\mathbb{Z}[X], \mathbf{S})$.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 1st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, thanks for fixing that. I had put the statement the way Connes said it in his talk, but evidently that was without some fine print.

   Ah, thanks for fixing that. I had put the statement the way Connes said it in his talk, but evidently that was without some fine print.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 1st 2010
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded that we can form many examples of hyperrings by quotienting a ring $R$ by a subgroup $G \subset R^{\times}$ of its multiplicative group.

   Added that we can form many examples of hyperrings by quotienting a ring $R$ by a subgroup $G \subset R^{\times}$ of its multiplicative group.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJul 14th 2010
   • (edited Jul 14th 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexJohn says [here](http://golem.ph.utexas.edu/category/2010/06/inevitability_in_mathematics.html#c033739) that the hypersemigroup is a monoid object in the category of sets and multivalued functions. Sounds the way I like, but I do not understand. I mean what is the monoidal product in the category of sets and multivalued functions (I guess a cartesian product, but what is its description in elementwise terms) ? Sounds like a student exercise but I would rather not blunder into possibly wrong attempts now.

   John says here that the hypersemigroup is a monoid object in the category of sets and multivalued functions. Sounds the way I like, but I do not understand. I mean what is the monoidal product in the category of sets and multivalued functions (I guess a cartesian product, but what is its description in elementwise terms) ? Sounds like a student exercise but I would rather not blunder into possibly wrong attempts now.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJul 15th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think the monoidal structure is the ordinary cartesian product of sets, which is however not the cartesian product in the category of sets and multivalued functions.

   I think the monoidal structure is the ordinary cartesian product of sets, which is however not the cartesian product in the category of sets and multivalued functions.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 15th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIt's probably high time that all these notions got sussed out properly, because certain aspects of [[hypergroup]] and [[hyperring]] as so far presented have an ad hoc look to them. These could probably be clarified by placing them in the context of the [[cartesian bicategory]] of relations and the cartesian monoidal subbicategory of functions. But to answer the immediate question: a multivalued function from $A$ to $B$ is just a relation from $A$ to $B$, and we are working in the bicategory of sets and relations. The monoidal structure is given by cartesian product of sets as you surmise, and the tensor product of relations $R: A \to B$, $S: C \to D$ is the relation $R \times S: A \times C \to B \times D$ defined by $$(R \times S)((a, c), (b, d)) = R(a, b) \wedge S(c, d)$$ This is a monoidal product (in the world of poset-enriched categories, or of 2-categories if you prefer), but not a cartesian monoidal product. Of course, we can define monoids in any monoidal category, so John's formulation makes sense. (Apparently the hypergroup experts intend "multivalued" to mean "at least one value", so here we are talking about *total* relations. But that's okay: it's clear that the monoidal product as defined above restricts to the bicategory of total relations.) Another way of thinking about all this is that we're working in the full (poset-enriched) subcategory of the monoidal category of sup-lattices and cocontinuous maps whose objects are power sets. A monoid in the monoidal category of sup-lattices is usually called a (noncommutative) [[quantale]], so John's definition boils down to a quantale structure on a power object, except that we must restrict to total relations. The funny thing is: I can't seem to rummage up many interesting examples of *total* power quantales off the top of my head! There seems to be a paucity of examples over at [[hyperring]], and it leads me to wonder what can be said about the finite examples. Can these be classified, or if not, can be easily produce a rich supply of them?

   It’s probably high time that all these notions got sussed out properly, because certain aspects of hypergroup and hyperring as so far presented have an ad hoc look to them. These could probably be clarified by placing them in the context of the cartesian bicategory of relations and the cartesian monoidal subbicategory of functions.

   But to answer the immediate question: a multivalued function from $A$ to $B$ is just a relation from $A$ to $B$, and we are working in the bicategory of sets and relations. The monoidal structure is given by cartesian product of sets as you surmise, and the tensor product of relations $R: A \to B$, $S: C \to D$ is the relation $R \times S: A \times C \to B \times D$ defined by

   $$(R \times S)((a, c), (b, d)) = R(a, b) \wedge S(c, d)$$

   This is a monoidal product (in the world of poset-enriched categories, or of 2-categories if you prefer), but not a cartesian monoidal product. Of course, we can define monoids in any monoidal category, so John’s formulation makes sense.

   (Apparently the hypergroup experts intend “multivalued” to mean “at least one value”, so here we are talking about total relations. But that’s okay: it’s clear that the monoidal product as defined above restricts to the bicategory of total relations.)

   Another way of thinking about all this is that we’re working in the full (poset-enriched) subcategory of the monoidal category of sup-lattices and cocontinuous maps whose objects are power sets. A monoid in the monoidal category of sup-lattices is usually called a (noncommutative) quantale, so John’s definition boils down to a quantale structure on a power object, except that we must restrict to total relations.

   The funny thing is: I can’t seem to rummage up many interesting examples of total power quantales off the top of my head! There seems to be a paucity of examples over at hyperring, and it leads me to wonder what can be said about the finite examples. Can these be classified, or if not, can be easily produce a rich supply of them?

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 15th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexActually, now that I think further, I believe one interesting example would be the Boolean Hecke algebra of a Coxeter group, which Jim Dolan and I were discussing a few years back. This would be worth putting on the Lab.

   Actually, now that I think further, I believe one interesting example would be the Boolean Hecke algebra of a Coxeter group, which Jim Dolan and I were discussing a few years back. This would be worth putting on the Lab.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 15th 2010
   • (edited Jul 15th 2010)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexHm, there is no extant entry for [[hypersemigroup]] or [[hypermonoid]], and it's not clear we want or need to create one. So I'll just record the example here for now. Let $W$ be a finite Coxeter group. A Coxeter group is really a group presentation, with a collection of involutive generators $s_1, \ldots, s_n$ and whose relations are of the form $(s_i s_j)^{m_{i j} = 1$, where $m_{i j}$ is a collection of integers specified by a Coxeter diagram or Coxeter matrix (assumed to be symmetric and with 1's down the diagonal). We may rewrite the presentation as $$s_{i}^{2} = 1, \qquad s_i s_j \ldots = s_j s_i \ldots$$ where the words on either side of the second equation alternate between $s_i$ and $s_j$ (for distinct $i$ and $j$) and have length $m_{i j}$. The Boolean Hecke algebra of a Coxeter group $W$ is a quantale whose underlying sup-lattice is $P W$, and whose quantalic multiplication is uniquely determined by a hypermonoid structure which satisfies $$s_{i}^{2} = \{s_i, 1\}$$ and described roughly as follows: given two reduced words $w_1$, $w_2$ in the Coxeter group representation, put the word $w_1 w_2$ into reduced form, resolving any occurrences of $s_{i}^2$ that occur according to the equation above. Now group elements $g \in W$ are equivalence classes of reduced words, and the recipe above actually gives a well-defined map $W \times W \to P(W)$ that gives a hypermonoid structure. The way I just portrayed it makes it seem very combinatorial and dependent on the theory of rewriting systems, but is can be made more conceptual if the Coxeter group $W$ is seen as coming from a BN-pair. How this usually works is that "B" is a Borel subgroup of a simple algebraic group $G$ and "N" is the normalizer of a maximal torus. The basic idea then is that the Boolean Hecke algebra is the algebra consisting of $G$-equivariant Boolean algebra maps $$P(G/B) \to P(G/B)$$ which as a sup-lattice is isomorphic to $P(B\backslash G/B)$, the power set of the set of double cosets. (Since such equivariant maps compose, this imparts a quantale structure to $P(B\backslash G/B)$.) When I refer to $W$ "coming from" a BN-pair, I mean according to some general theory that these double cosets are in bijection with the elements of some Coxeter group $W$. Hence $P(W) \cong P(B\backslash G/B)$, but the algebra structure on $P(B \backslash G/B)$ can be regarded as a deformation of the group algebra structure. This example of a hypermonoid deserves to be discussed at much greater length, but not necessarily under [[hypermonoid]].

   Hm, there is no extant entry for hypersemigroup or hypermonoid, and it’s not clear we want or need to create one. So I’ll just record the example here for now.

   Let $W$ be a finite Coxeter group. A Coxeter group is really a group presentation, with a collection of involutive generators $s_1, \ldots, s_n$ and whose relations are of the form , where $m_{i j}$ is a collection of integers specified by a Coxeter diagram or Coxeter matrix (assumed to be symmetric and with 1’s down the diagonal). We may rewrite the presentation as

   $$s_{i}^{2} = 1, \qquad s_i s_j \ldots = s_j s_i \ldots$$

   where the words on either side of the second equation alternate between $s_i$ and $s_j$ (for distinct $i$ and $j$) and have length $m_{i j}$.

   The Boolean Hecke algebra of a Coxeter group $W$ is a quantale whose underlying sup-lattice is $P W$, and whose quantalic multiplication is uniquely determined by a hypermonoid structure which satisfies

   $$s_{i}^{2} = \{s_i, 1\}$$

   and described roughly as follows: given two reduced words $w_1$, $w_2$ in the Coxeter group representation, put the word $w_1 w_2$ into reduced form, resolving any occurrences of $s_{i}^2$ that occur according to the equation above. Now group elements $g \in W$ are equivalence classes of reduced words, and the recipe above actually gives a well-defined map $W \times W \to P(W)$ that gives a hypermonoid structure.

   The way I just portrayed it makes it seem very combinatorial and dependent on the theory of rewriting systems, but is can be made more conceptual if the Coxeter group $W$ is seen as coming from a BN-pair. How this usually works is that “B” is a Borel subgroup of a simple algebraic group $G$ and “N” is the normalizer of a maximal torus. The basic idea then is that the Boolean Hecke algebra is the algebra consisting of $G$-equivariant Boolean algebra maps

   $$P(G/B) \to P(G/B)$$

   which as a sup-lattice is isomorphic to $P(B\backslash G/B)$, the power set of the set of double cosets. (Since such equivariant maps compose, this imparts a quantale structure to $P(B\backslash G/B)$.) When I refer to $W$ “coming from” a BN-pair, I mean according to some general theory that these double cosets are in bijection with the elements of some Coxeter group $W$. Hence $P(W) \cong P(B\backslash G/B)$, but the algebra structure on $P(B \backslash G/B)$ can be regarded as a deformation of the group algebra structure.

   This example of a hypermonoid deserves to be discussed at much greater length, but not necessarily under hypermonoid.

10. • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeJul 15th 2010
    • PermaLink
    Author: zskoda
    Format: MarkdownItexGreat answers! My personal opinion is that long term it would be good to have separate entries [[hypermonoid]], [[hypersemigroup]], [[hyperfield]] in addition to the present [[hyperring]]. It seems that the subject is becoming important.

    Great answers! My personal opinion is that long term it would be good to have separate entries hypermonoid, hypersemigroup, hyperfield in addition to the present hyperring. It seems that the subject is becoming important.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItex> This example of a hypermonoid deserves to be discussed at much greater length, but not necessarily under hypermonoid. Just please don't forget to put it _somewhere_ on the $n$Lab. Why not at [[hypermonoid]]? You can still move it later.

    This example of a hypermonoid deserves to be discussed at much greater length, but not necessarily under hypermonoid.

    Just please don’t forget to put it somewhere on the $n$Lab. Why not at hypermonoid? You can still move it later.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 15th 2010
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexOkay, I just did that Urs.

    Okay, I just did that Urs.

13. • CommentRowNumber13.
    • CommentAuthorvarkor
    • CommentTimeAug 23rd 2024
    • PermaLink
    Author: varkor
    Format: MarkdownItexAdded related references. <a href="https://ncatlab.org/nlab/revision/diff/hyperring/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/14">v14</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    Added related references.

    diff, v14, current

14. • CommentRowNumber14.
    • CommentAuthorJohn Baez
    • CommentTimeNov 2nd 2024
    • PermaLink
    Author: John Baez
    Format: MarkdownItexWhy does the definition of 'hyperring' here demand $0 \neq 1$? That seems like a bad idea: for example, it means not every ring is a hyperring, and it prevents there from being a terminal hyperring. Can we just get rid of this? <a href="https://ncatlab.org/nlab/revision/diff/hyperring/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/15">v15</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    Why does the definition of ’hyperring’ here demand $0 \neq 1$? That seems like a bad idea: for example, it means not every ring is a hyperring, and it prevents there from being a terminal hyperring. Can we just get rid of this?

    diff, v15, current

15. • CommentRowNumber15.
    • CommentAuthorJohn Baez
    • CommentTimeNov 2nd 2024
    • PermaLink
    Author: John Baez
    Format: MarkdownItexI couldn't resist eliminating the rule $0 \ne 1$, and I also explained some examples of hyperrings more carefully. <a href="https://ncatlab.org/nlab/revision/diff/hyperring/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/15">v15</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    I couldn’t resist eliminating the rule $0 \ne 1$, and I also explained some examples of hyperrings more carefully.

    diff, v15, current

16. • CommentRowNumber16.
    • CommentAuthorMadeleine Birchfield
    • CommentTimeNov 2nd 2024
    • (edited Nov 2nd 2024)
    • PermaLink
    Author: Madeleine Birchfield
    Format: MarkdownItexHyperrings are defined in the references listed on the page as having the $0 \neq 1$ condition; i.e. see [Connes & Consani](https://arxiv.org/abs/1001.4260). Do you have a reference which states that hyperrings can be trivial?

    Hyperrings are defined in the references listed on the page as having the $0 \neq 1$ condition; i.e. see Connes & Consani.

    Do you have a reference which states that hyperrings can be trivial?

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2024
    • (edited Nov 3rd 2024)
    • PermaLink
    Author: Urs
    Format: MarkdownItexGenerally good style not to silently delete conditions, better to add a remark. (Otherwise, next somebody comes and silently adds the condition back in, thinking it's missing.) I have now dug out and listed four references ([here](https://ncatlab.org/nlab/show/hyperring#ReferencesForTheDefinition)) two of which include $0 \neq 1$ (the algebraic geometers) and two of which don't (reviewers of Krasner's original theory). I couldn't easily find copies of the original references, though. If anyone has them, let's (add the links and) check what conditions are stated there. Besides this I also made a bunch of cosmetic edits: Touched wording and hyperlinking of the first paragraphs, added the context menu "Algebra", added more references and more reference data (doi etc) and re-organized them somewhat to make the list more systematic. Also replaced all the clunky HTML-encoding of French accents with plain symbols. <a href="https://ncatlab.org/nlab/revision/diff/hyperring/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/17">v17</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    Generally good style not to silently delete conditions, better to add a remark. (Otherwise, next somebody comes and silently adds the condition back in, thinking it’s missing.)

    I have now dug out and listed four references (here) two of which include $0 \neq 1$ (the algebraic geometers) and two of which don’t (reviewers of Krasner’s original theory).

    I couldn’t easily find copies of the original references, though. If anyone has them, let’s (add the links and) check what conditions are stated there.

    Besides this I also made a bunch of cosmetic edits:

    Touched wording and hyperlinking of the first paragraphs, added the context menu “Algebra”, added more references and more reference data (doi etc) and re-organized them somewhat to make the list more systematic. Also replaced all the clunky HTML-encoding of French accents with plain symbols.

    diff, v17, current

18. • CommentRowNumber18.
    • CommentAuthorJohn Baez
    • CommentTimeNov 4th 2024
    • PermaLink
    Author: John Baez
    Format: MarkdownItexI decided we should impose $0 \ne 1$ for a hyperfield, though not for a general hyperring. <a href="https://ncatlab.org/nlab/revision/diff/hyperring/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/18">v18</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    I decided we should impose $0 \ne 1$ for a hyperfield, though not for a general hyperring.

    diff, v18, current

19. • CommentRowNumber19.
    • CommentAuthorJohn Baez
    • CommentTimeNov 5th 2024
    • PermaLink
    Author: John Baez
    Format: MarkdownItexNoticed that a condition on the group $G \subseteq R^\times$ is necessary in defining quotient hyperrings; this is in Nakassis' book. <a href="https://ncatlab.org/nlab/revision/diff/hyperring/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/19">v19</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    Noticed that a condition on the group $G \subseteq R^\times$ is necessary in defining quotient hyperrings; this is in Nakassis’ book.

    diff, v19, current

20. • CommentRowNumber20.
    • CommentAuthorJohn Baez
    • CommentTimeNov 5th 2024
    • PermaLink
    Author: John Baez
    Format: MarkdownItexAdded quote by Viro in the "Idea" section. I added a reference to the paper this quote is from, near the very bottom of the references, but I forget how to link to that reference. <a href="https://ncatlab.org/nlab/revision/diff/hyperring/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/20">v20</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    Added quote by Viro in the “Idea” section. I added a reference to the paper this quote is from, near the very bottom of the references, but I forget how to link to that reference.

    diff, v20, current

21. • CommentRowNumber21.
    • CommentAuthorJohn Baez
    • CommentTimeNov 5th 2024
    • PermaLink
    Author: John Baez
    Format: MarkdownItexIntroduced Viro's example of the triangle hyperfield, introduced the 'double distributive law', and pointed out that the triangle hyperfield does not obey this law, while every hyperring obtained by the quotient construction from a ring does obey this law. <a href="https://ncatlab.org/nlab/revision/diff/hyperring/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/21">v21</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    Introduced Viro’s example of the triangle hyperfield, introduced the ’double distributive law’, and pointed out that the triangle hyperfield does not obey this law, while every hyperring obtained by the quotient construction from a ring does obey this law.

    diff, v21, current

22. • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2024
    • (edited Nov 6th 2024)
    • PermaLink
    Author: Urs
    Format: MarkdownItexre [#20](#Comment_119686): Here how to equip the bibitem with an "anchor": * {#Viro10} Oleg Viro: *Hyperfields for... and here how to refer to it from the main text: From [Viro 2010](#Viro10):... <a href="https://ncatlab.org/nlab/revision/diff/hyperring/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/23">v23</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    re #20:

    Here how to equip the bibitem with an “anchor”:

      * {#Viro10} Oleg Viro: *Hyperfields for...
    

    and here how to refer to it from the main text:

      From [Viro 2010](#Viro10):...
    

    diff, v23, current

23. • CommentRowNumber23.
    • CommentAuthorRodMcGuire
    • CommentTimeNov 6th 2024
    • PermaLink
    Author: RodMcGuire
    Format: MarkdownItexApparently "D. Stratigopoulos" (1969) is "Dimitrios Stratigopoulos" and Google finds a few other articles under that name though here is a 1988 Journal of Number Theory article where his name is "Dimitri" and lists his affiliation as Patras University in Greece. [vol 28 issue 1](https://www.sciencedirect.com/journal/journal-of-number-theory/vol/28/issue/1) <a href="https://ncatlab.org/nlab/revision/diff/hyperring/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/hyperring/24">v24</a>, <a href="https://ncatlab.org/nlab/show/hyperring">current</a>

    Apparently “D. Stratigopoulos” (1969) is “Dimitrios Stratigopoulos” and Google finds a few other articles under that name though here is a 1988 Journal of Number Theory article where his name is “Dimitri” and lists his affiliation as Patras University in Greece.

    vol 28 issue 1

    diff, v24, current