Fully simple semihypergroups

Abstract

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]

Semihypergroups obtained by merging of 0-semigroups with groups

“…The relations β, β * are called fundamental relations on H [1,9,13,14,18]. The interested reader can find all relevant definitions, many properties and applications of fundamental relations, even in more abstract contexts, also in [4,5,7,8,10,15].…”

On hypercyclic fully zero-simple semihypergroups

“…is a simple hypercyclic semihypergroup. By Theorem 3.1 in [11], the relation β H+ is transitive. Proof By Theorem 3, we have that a ̸ = 0 and c ̸ = 0 ; otherwise, (a, c) ∈ β.…”

On Hypergroups with a β-Class of Finite Height

“…Certain equivalence relations, called fundamental relations, introduce natural correspondences between algebraic hyperstructures and classical algebraic structures. These equivalence relations have the property of being the smallest strongly regular equivalence relations such that the corresponding quotients are classical algebraic structures [4][5][6][7][8][9][10][11]. For example, if (H, •) is a hypergroup, then the fundamental relation β is transitive [12][13][14] and the quotient set H/β is a group.…”