Results 1  10
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14
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 Part I
"... In this contribution we construct noncommutative transposition hypergroups of integral operators on spaces of continuous functions which are created by Fredholm integral equations of the first and second kinds. Moreover, we investigate the obtained hyperstructures as transposition hypergroups and al ..."
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In this contribution we construct noncommutative transposition hypergroups of integral operators on spaces of continuous functions which are created by Fredholm integral equations of the first and second kinds. Moreover, we investigate the obtained hyperstructures as transposition hypergroups and also related quasihypergroups of blocks of equivalence of integral operators. Theory of linear integral equations which are a certain continuous analogue of systems of linear algebraic equations belongs to classical parts of contemporary pure and applied mathematics and plays an important role from the point of view of technical sciences. The basic constructed structures are ordered groups of integral operators. Moreover, we use also the object function (where the corresponding binary hyperoperation on an ordered group is defined as principal end generated by products of pairs of elements of the considered group) of a functor enabling the transfer from the category of ordered groups and their isotone homomorphisms into the category of hypergroups and their inclusion homomorphisms. The basic group of integral operators contains an invariant subgroup. Using another binary operation on the set of suitable Fredholm integral operators of the second kind we get a group with a significant noninvariant subgroup of operators of the first kind enabling to construct a quasihypergroup of decomposition classes of operators, structure of which is also clarified.
Multivalued tNorms and tConorms
, 2003
"... We present a procedure for constructing multivalued tnorms and tconorms. Our construction uses a pair of singlevalued tnorms and the pair of dual tconorms to construct intervalvalued tnorms and tconorms #. ..."
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We present a procedure for constructing multivalued tnorms and tconorms. Our construction uses a pair of singlevalued tnorms and the pair of dual tconorms to construct intervalvalued tnorms and tconorms #.
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}.
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.
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.
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
Fuzzy Hyperideals of Fuzzy Hyperrings 1
"... Abstract: In this paper we introduce and analyze the notion of fuzzy hyperideal of a fuzzy hyperring. Then we investigate some properties based on homomorphisms between fuzzy hyperrings. Also, we try to extend the notion of cuts in fuzzy structures for fuzzy hyperrings. We study the quotient structu ..."
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Abstract: In this paper we introduce and analyze the notion of fuzzy hyperideal of a fuzzy hyperring. Then we investigate some properties based on homomorphisms between fuzzy hyperrings. Also, we try to extend the notion of cuts in fuzzy structures for fuzzy hyperrings. We study the quotient structure of a fuzzy hyperring and also we define fuzzy hypercongruences. Then we investigate some connections between hyperrings and fuzzy hyperrings using the notions of hypercongruences and fuzzy hypercongruences. We introduce and study induced equivalence relations on hyperrings and fuzzy hyperrings and investigate their properties. Finally, we define join hyperrings and fuzzy join hyperrings. Key words: Fuzzy hyperring • fuzzy hyperideal • quotient hyperring • fuzzy hypercongruence • join hyperrings
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.