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24
THE HYPERRING OF ADÈLE CLASSES
"... Abstract. We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK = AK/K × of a global field K. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G ⊂ ..."
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Abstract. We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK = AK/K × of a global field K. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G ⊂ R × is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G ∪ {0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semilinear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous nonDesarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the
Transposition hypergroups of Fredholm integral operators and related hyperstructures
 AHA 2005, Babolsar, Iran
"... ..."
Fuzzy sets and non complete 1hypergroups
 An. St. Univ. Ovidius Constanta
, 2005
"... In this paper it has been studied the sequence of membership functions and of join spaces (see [7]), determined by a class of 1–hypergroups which are not complete (see [2]). ..."
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In this paper it has been studied the sequence of membership functions and of join spaces (see [7]), determined by a class of 1–hypergroups which are not complete (see [2]).
The Hyperrings of Order 3
 JOURNAL OF INTEGER SEQUENCES, VOL. 11 (2008), ARTICLE 08.3.2
, 2008
"... We first explain the historical and logical relations of hyperstructures introduced by M. Krasner and R. Rota, and generalized by T. Vougiouklis. Then, with our new algorithm based on our previous results on hypergroups and Hvgroups of order 2, 3 and 4, we enumerate hyperrings and Hvrings. More pr ..."
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We first explain the historical and logical relations of hyperstructures introduced by M. Krasner and R. Rota, and generalized by T. Vougiouklis. Then, with our new algorithm based on our previous results on hypergroups and Hvgroups of order 2, 3 and 4, we enumerate hyperrings and Hvrings. More precisely, we found 63 hyperrings of order 2, 875 Hvrings of order 2 and 33,277,642 hyperrings of order 3. Finally, in this new context, we study a new connection between groups and hypergroups via the notion of duality.
Dépt. de mathématiques et de statistique,
, 2008
"... In 1986, the second author classified the minimal clones on a finite universe into five types. We extend this classification to infinite universes and to multiclones. We show that every nontrivial clone contains a ”small ” clone of one of the five types. From it we deduce, in part, an earlier resul ..."
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In 1986, the second author classified the minimal clones on a finite universe into five types. We extend this classification to infinite universes and to multiclones. We show that every nontrivial clone contains a ”small ” clone of one of the five types. From it we deduce, in part, an earlier result, namely that if C is a clone on a universe A with at least two elements, that contains all constant operations, then all binary idempotent operations are projections and some mary idempotent operation is not a projection some m ≥ 3 if and only if there is a Boolean group G on A for which C is the set of all operations f(x1,..., xn) of the form a + ∑ i∈I xi for a ∈ A and I ⊆ {1,...,n}.
Enumeration of Varlet and Comer hypergroups
"... In this paper, we study hypergroups determined by lattices introduced by Varlet and Comer, especially we enumerate Varlet and Comer hypergroups of orders less than 50 and 13, respectively. 1 Basic definitions and results An algebraic hyperstructure is a natural generalization of a classical algebrai ..."
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In this paper, we study hypergroups determined by lattices introduced by Varlet and Comer, especially we enumerate Varlet and Comer hypergroups of orders less than 50 and 13, respectively. 1 Basic definitions and results An algebraic hyperstructure is a natural generalization of a classical algebraic structure. More precisely, an algebraic hyperstructure is a nonempty set H endowed with one or more hyperoperations that associate with two elements of H not an element, as in a classical structure, but a subset of H. One of the interests of the researchers in the field of hyperstructures is to construct new hyperoperations using graphs [18], binary relations
STRONGLY TRANSITIVE GEOMETRIC SPACES: APPLICATIONS TO HYPERRINGS
"... Abstract. In this paper, we determine two families R and G of subsets of a hyperring R and sufficient conditions such that two geometric spaces (R,R) and (R,G) are strongly transitive. Moreover, we prove that the relations Γ and α are strongly regular equivalence relations on a hyperfield or a hyper ..."
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Abstract. In this paper, we determine two families R and G of subsets of a hyperring R and sufficient conditions such that two geometric spaces (R,R) and (R,G) are strongly transitive. Moreover, we prove that the relations Γ and α are strongly regular equivalence relations on a hyperfield or a hyperring such that (R,+) has an identity element. 1.
ACTA UNIVERSITATIS APULENSIS No 15/2008 PRODUCTS OF MULTIALGEBRAS AND THEIR FUNDAMENTAL ALGEBRAS
"... Abstract. An important tool in the hyperstructure theory is the fundamental relation. The factorization of a multialgebra modulo its fundamental relation provides a functor into the category of universal algebras. The question that lead us to the results we will present is whether this functor commu ..."
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Abstract. An important tool in the hyperstructure theory is the fundamental relation. The factorization of a multialgebra modulo its fundamental relation provides a functor into the category of universal algebras. The question that lead us to the results we will present is whether this functor commutes with the products. 2000 Mathematics Subject Classification: 08A05, 20N20, 08A99. 1.
Structures of N∧Hyperideals in Left Almost ∧Semihypergroups 1
"... Abstract: In this paper, the notions of sub NLAΓsemihypergroup, NΓhyperideal and NbiΓhyperideal in left almost Γsemihypergroup are introduced and several properties are investigated. ..."
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Abstract: In this paper, the notions of sub NLAΓsemihypergroup, NΓhyperideal and NbiΓhyperideal in left almost Γsemihypergroup are introduced and several properties are investigated.