1,720,964 research outputs found
Constructing Steiner triple systems partially embedded in a projective plane
No full textThe usefulness of block designs in many natural and social sciences has been long astablished. In recent years, a considerable effort has been performed toward the construction of block designs. It would be of interest to construct designs using the geometric structure of a finite projective plane. In this paper by a well-known construction of Hanani connected with the idea of a Group Divisible Design we give some triplications of Steiner triple systems obtained from some simple GDD embedded in a finite projective plane
Fully simple semihypergroups
No full textIn this paper we consider the class of semihypergroups H such that all subsemihypergroups K ⊆ H are simple and, when |K| ≥ 3 the fundamental relation β_K is not transitive. For these semihypergroups we prove that hyperproducts of elements in H have size ≤ 2 and the quotient semigroup H/β∗ is trivial. This last result allows us to completely characterize these semihypergroups in terms of a small set of simple semihypergroups of size 3. Finally, we solve a problem on strongly simple semihypergroups introduced in [11]
Commutativity and Completeness Degrees of Weakly Complete Hypergroups
No full textWe introduce a family of hypergroups, called weakly complete, generalizing the construction of complete hypergroups. Starting from a given group G, our construction prescribes the β-classes of the hypergroups and allows some hyperproducts not to be complete parts, based on a suitably defined relation over G. The commutativity degree of weakly complete hypergroups can be related to that of the underlying group. Furthermore, in analogy to the degree of commutativity, we introduce the degree of completeness of finite hypergroups and analyze this degree for weakly complete hypergroups in terms of their β-classes
1-hypergroups of small sizes
Full text linkIn this paper, we show a new construction of hypergroups that, under appropriate conditions, are complete hypergroups or non-complete 1-hypergroups. Furthermore, we classify the 1-hypergroups of size 5 and 6 based on the partition induced by the fundamental relation β. Many of these hypergroups can be obtained using the aforesaid hypergroup construction
Hypergroups with a strongly unilateral identity
No full textAmong hyperstructures of type U on the right having small size, the order 6 is a relevant case. Indeed, only if the order is \leq 6 there exist proper semihypergrops and hypergroups of type U on the right whose right scalar identity is not also left identity. In the present paper we show a construction of hypergroups of type U on the right whose right scalar identity is not also left identity. That construction characterizes completely the case of order 6, and allows to introduce a semi-ordering structure within that case. With the help of that semi-ordering, and of symbolic computation software, we show that these hypergroups can be obtained as hyperproduct extensions of 41 minimal hypergroups, and that the number of their isomorphism classes is 946
On strongly conjugable extensions of hypergroups of type U with scalar identity
No full textLet S_n denote the class of hypergroups of type U on the right of size n with bilateral scalar identity. In this paper we consider the hypergroups (H,◦) ∈ S_7 which own a proper and non-trivial subhypergroup h. For these hypergroups we prove that h is closed if and only if
(H − h) ◦ (H − h) = h. Moreover we consider the set E_7 of hypergroups in S_7 that own the above property. On this set, we introduce a partial ordering induced by the inclusion of hyperproducts. This partial ordering allows us to give a complete characterization of hypergroups in E_7 on the basis of a small set of minimal hypergroups, up to isomorphisms. This analysis gives a partial answer to a problem raised in [5] concerning the existence in S_n of proper hypergroups having singletons as special hyperproducts
On the hypergroups with four proper pairs and three or four non-scalar elements
No full textIn this paper the authors studi the hypergroups, such that are exactly four pairs of elements, which define hyperproducts (i.e. with P(W)04), and such that there are three or four non-scalar elements. They solve the combinatorial problem of finding, up to isomorphism, all the tables of the aforesaid hypergroups and therefore, counting also on the papers [3], [4], they bring to a close the problem of finding all the finite hypegroups such that |P(H)|=4
Hypergroups of type U on the right of size five. Part two
No full textThe hypergroups H of type U on the right can be classified in terms of the family P_1={ 1ox | x\in H}, where 1\in H is the right scalar identity. If the size of H is 5, then P_1 can assume only 6 possible values, three of which have been studied in [3]. In this paper, we completely describe other two of the remaining possible cases: a) P_1 ={{1}, {2, 3}, {4}, {5}}; b) P_1 ={{1}, {2,3}, {4, 5}}. In these cases, P_1 is a partition of H and the equivalence relation associated to it is a regular equivalence on H. We find that, apart of isomorphisms, there are exactly 41 hypergroups in case a), and 56 hypergroup in case b)
On the hypergroups with four proper pairs and without scalars
No full textIn this paper the authors study the hypergroups such that there are exactly four pairs of elements, which define proper hyperproducts and such that there are no scalars. they solve the combinatorial problem of finding, up to isomorphism, all the tables of the aforesaid hypergroups
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