Fully simple semihypergroups, transitive digraphs, and sequence A000712

Abstract

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

On path hypercompositions in graphs and automata

“…An automaton A is a collection of five objects [24] and others studied hypergroups associated with graphs. G. G. Massouros [14][15][16][17][18] and after him J. Chvalina [1] studied hypergroups associated with automata.…”

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.…”

“…Our main aim is to deepen the understanding of the properties of the fundamental relation β in hypergroups and to enumerate the non-isomorphic hypergroups fulfilling certain conditions on the cardinality of the hearth. This task belongs to an established research field that deals with fundamental relations and enumerative problems in hypercompositional algebra [5,6,[13][14][15]. The plan of this article is the following: After introducing some basic definitions and notations to be used throughout this article, in Section 3, we define the notion of height h(β(x)) of an equivalence class β(x).…”

Isomorphism Theorems in the Primary Categories of Krasner Hypermodules

“…A hypergroup is basically a set endowed by an associative multivalued binary operation, which fulfills an additional condition called reproducibility. Their inspection may reveal complex relationships among algebra, combinatorics, graphs, and numeric sequences [6,7]. Indeed, hyperstructures are inherently more complicated and bizarre than their classical counterparts.…”