In 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]
Fully simple semihypergroups have been introduced in [9], motivated by the study of the transitivity of the fundamental relation β in semihypergroups. Here, we determine a transver- sal of isomorphism classes of fully simple semihypergroups with a right absorbing element. The structure of that transversal can be described by means of certain transitive, acyclic digraphs. Moreover, we prove that, if n is an integer ≥2, then the number of isomorphism classes of fully simple semihypergroups of size n + 1, with a right absorbing element, is the (n + 1)-th term of sequence A000712 in [20], namely, nk=0 p(k)p(n − k), where p(k) denotes the number of nonincreasing partitions of integer k
In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β -class and we analyze properties of hypergroups where the height of a β -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one β -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is n ≤ 4 , apart from isomorphisms.
In 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.
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