Narrow Results

On a conformal net ${𝒜}$, one can consider collections of unital completely positive maps on each local algebra ${𝒜}(I)$, subject to natural compatibility, vacuum preserving and conformal covariance conditions. We call quantum operations on ${𝒜}$ the subset of extreme such maps. The usual automorphisms of ${𝒜}$ (the vacuum preserving invertible unital *-algebra morphisms) are examples of quantum operations, and we show that the fixed point subnet of ${𝒜}$ under all quantum operations is the Virasoro net generated by the stress-energy tensor of ${𝒜}$. Furthermore, we show that every irreducible conformal subnet ${ℬ}\subset {𝒜}$ is the fixed points under a subset of quantum operations.

When ${ℬ}\subset {𝒜}$ is discrete (or with finite Jones index), we show that the set of quantum operations on ${𝒜}$ that leave ${ℬ}$ elementwise fixed has naturally the structure of a compact (or finite) hypergroup, thus extending some results of [M. Bischoff, Generalized orbifold construction for conformal nets, Rev. Math. Phys. 29 (2017) 1750002]. Under the same assumptions, we provide a Galois correspondence between intermediate conformal nets and closed subhypergroups. In particular, we show that intermediate conformal nets are in one-to-one correspondence with intermediate subfactors, extending a result of Longo in the finite index/completely rational conformal net setting [R. Longo, Conformal subnets and intermediate subfactors, Comm. Math. Phys. 237 (2003) 7–30].

Let $H$ be a commutative hypergroup and $L$ a discrete commutative hypergroup. In this paper we introduce a commutative hypergroup ${𝒦}(H,\phi ,L)$ associated with a hyperfield $\phi$ of $H$ based on $L$. Moreover, for the hyperfield $\phi$ of a compact commutative hypergroup $H$ of strong type based on a discrete commutative hypergroup $L$ of strong type, we introduce the dual hyperfield $\widehat{\phi }$ of $\widehat{L}$ based on $Ĥ$ and show that $\widehat{{𝒦}}(H,\phi ,L)\cong {𝒦}(\widehat{L},\widehat{\phi },Ĥ)$.

We prove that all hypergroups of order four are commutative and that there exists a non-comutative hypergroup of order five. These facts imply that the minimum order of non-commutative hypergroups is five, even though the minimum order of non-commutative groups is six.

In this paper, we define an equivalence relation induced by $n$-ary subpolygroups and show that such relation is full conjugation when $n$-ary subpolygroup is normal. Then, we introduce a construction and some properties of direct product of $n$-ary polygroups via $n$-ary factor polygroups. This construction in principle is enlargement of elements of $n$-ary polygroup except zero. Finally, we introduce and investigate the semidirect product of $n$-ary polygroups via $n$-ary factor polygroups.

A zero forcing set is a new concept in Graph Theory which was introduced in recent years. In this paper, we investigate the relationship between zero forcing sets and algebraic hyperstructures. To this end, we present some new definitions by considering a zero forcing process on a graph $G$. These definitions help us analyze the zero forcing process better and construct various hypergroups and join spaces on the vertex set of graph $G$. Finally, we give some examples to clarify these hyperstructures.

In this paper, we determine a family ${𝔓}_{\Re }^H$ of subsets of a hypergroup $H$ such that the geometric space $(H,{𝔓}_{\Re }^H)$ is strongly transitive. We show that if $(H,\circ )$ is a hypergroup and $\Re$ is strongly regular relation then ${\rho }_{\Re }$ is an equivalence relation.

Hypergroup is the generalized concept of group introduced first by Marty. Since then, the study of it and its applications has been of great importance. This paper deals with representations of the hypergroup associated to the braid group $B_n$. First, we define our hypergroup $(B_n,\star )$ and our semihyperring $(P,\oplus ,\odot )$ over the set of non-negative integers. Next, we construct non-trivial matrix representations of $B_n$ of degree $k$, for all $k\in \mathbb{N}$. Finally, we find all matrix representations of $(B_n,\star )$ with degree less than $3$ and classify them according to irreducibility and unitarizability.

In this paper, we study a wide class of $H_v$-modules obtained from ordinary $R$-modules by using $P$-hyperoperations. We define a new $P$-$H_v$-module over $P$-$H_v$-rings and study several properties of it starting from the isomorphism theorem to the fundamental relation.

This paper presents a general framework for the study of the relations between hypergraphs and hypergroups based on approximation operators. Indeed, by a given hypergraph $\mathrm{\Gamma }$, an equivalence relation $\rho$ on the set of vertices and using the notion of lower and upper approximations, we investigate two new hyperoperations on the set of vertices. In particular, by considering certain conditions on the equivalence relation $\rho$, we obtain two related $H_v$-groups and we prove some results in this respect.

Cyclic hypergroups are of great importance due to their applications to many fields in mathematics. In this paper, we classify all commutative single power cyclic hypergroups of order three and period two. Moreover, we prove some new interesting properties regarding cyclic hypergroups.

In this paper, we consider polygroup $〈P,\circ ,1{,}^{-1}〉$ and prove necessary and sufficient conditions such that $P$ is non-commutative. Then by using the Maple programming, we obtain all polygroups of order less than five up to isomorphism. In fact, we determine all 115 non-isomorphic polygroups of order less than five and characterize them by their fundamental groups, i.e., polygroups with same fundamental group, say $G$, classifies in the class $[G]$. Finally, we obtain that the fundamental groups of 94 polygroups are the trivial group. The numbers of polygroups in classes $[{\mathbb{Z}}_2]$ and $[{\mathbb{Z}}_3]$ are 16 and 3, respectively, and the classes $[{\mathbb{Z}}_2\times {\mathbb{Z}}_2]$ and $[{\mathbb{Z}}_4]$ are singleton.

The purpose of this paper is the study of hypergroups associated with hypergraphs. In this regard, we construct a hypergroupoid by defining a hyperoperation on the set of degrees of vertices of a hypergraph. We will investigate some conditions to have a hypergroup.

Cayley graph is a graph that encodes the abstract structure of a group. It gives a way of encoding information about a group in a graph. On the other hand, hypergroup is a generalization of group in which the composition of any two elements is a non-empty set. The purpose of this paper is to find a suitable generalization of Cayley graphs to cover hypergroups. More precisely, we introduce generalized Cayley graphs over hypergroups, study their properties and find a simple tool to construct large connected GCH-graphs from smaller $\mathrm{\text{GCH}}$-graphs by using graph product.