Fundamental relations in simple and 0-simple semihypergroups of small size

Abstract

We consider the fundamental relations β and γ in simple and 0-simple semihypergroups, especially in connection with certain minimal cardinality questions. In particular, we enumerate and exhibit all simple and 0-simple semihypergroups having order 3 where β is not transitive, apart of isomorphisms. Moreover, we show that the least order for which there exists a strongly simple semihypergroup where β is not transitive is 4. Finally, we prove that γ is transitive in all simple semihypergroups, and determine necessary and sufficient conditions for a 0-simple semihypergroup to have γ transitive. The latter results obviously hold also for simple and 0-simple semigroups. Mathematics Subject Classification (2010) 20N20 · 05A99

On Hypergroups with a β-Class of Finite Height

“…A subhypergroup K of a hypergroup (H, •) is said to be conjugable if it satisfies the following property: for all x ∈ H, there exists x ∈ H such that xx ⊆ K. The interested reader can find all relevant definitions, many properties, and applications of fundamental relations, even in more abstract contexts, also in [18][19][20][21][22][23][24][25][26][27][28].…”

Semihypergroups obtained by merging of 0-semigroups with groups

“…In our preceding papers [5,7,12] we faced the study of simple semihypergroups where the fundamental relation β is not transitive, in all subsemihypergroups of size ≥ 3. These semihypergroups, which we called fully simple, own a right (or left) zero scalar element and all their hyperproducts have size ≤ 2.…”

“…The six semihypergroups in Example 3.5 belong to the list of fourteen 0-semihypergroups of size 3 where the relation β is not transitive [12,Thm. 5.6].…”

“…Moreover, we show that these semihypergroups can be obtained as merging of a 0-semigroup with an elementary Abelian 2-group. Firstly, we borrow the following result from Theorem 5.6 in [12]. Proof.…”

On hypercyclic fully zero-simple semihypergroups

“…The semihypergroups are the simplest algebraic hyperstructures that possess the properties of closure and associativity. Some scholars have studied different aspects of semihypergroups [2,5,8,9,19,20,[22][23][24] and interesting problems arise in the study of their so-called fundamental relations [1,7,16,21,25], which leads to analyzing the conditions for their transitivity, and minimal cardinality problems. In [16] the authors found all simple and zero-simple semihypergroups of size 3 , such that the fundamental relation β is not transitive, apart from isomorphisms.…”