THE TRANSPOSITION AXIOM IN HYPERCOMPOSITIONAL STRUCTURES

The hypergroup (as defined by F. Marty), being a very general algebraic structure, was subsequently quickly enriched with additional axioms. One of these is the transposition axiom, the utilization of which led to the creation of join spaces (join hypergroups) and of transposition hypergroups. These hypergroups have numerous applications in geometry, formal languages, thetheory of automata and graph theory. This paper deals with transposition hypergroups. It also introduces the transposition axiom to weaker structures, which result from the hypergroup by the removal of certain axioms, thus defining the transposition hypergroupoid, the transposition semi-hypergroup and the transposition quasi-hypergroup. Finally, it presents hypercompositional structures with internal or external compositions and hypercompositions, in which the transposition axiom is valid. Such structures emerged during the study of formal languages and the theory of automata through the use of hypercompositional algebra

Enumeration of Rosenberg-type hypercompositional structures defined by binary relations

AbstractEvery binary relation ρ on a set H,(card(H)>1) can define a hypercomposition and thus endow H with a hypercompositional structure. In this paper, binary relations are represented by Boolean matrices. With their help, the hypercompositional structures (hypergroupoids, hypergroups, join hypergroups) that emerge with the use of the Rosenberg’s hyperoperation are characterized, constructed and enumerated using symbolic manipulation packages. Moreover, the hyperoperation given by x∘x={z∈H|(z,x)∈ρ} and x∘y=x∘x∪y∘y is introduced and connected to Rosenberg’s hyperoperation, which assigns to every (x,y) the set of all z such that either (x,z)∈ρ or (y,z)∈ρ

A family of 0-simple semihypergroups related to sequence A000070

For any integer n 65 2, let R_0(n + 1) be the class of 0-semihypergroups H of size n + 1 such that {y} 86 xy 86 {0, y} for all x, y 08 H - {0}, all subsemihypergroups K 86 H are 0-simple and, when |K| 65 3, the fundamental relation \u3b2_K is not transitive. We determine a transversal of isomorphism classes of semihypergroups in R0(n + 1) and we prove that its cardinality is the (n + 1)-th term of sequence A000070 in [21], namely, 11 _{k=0}^n p(k), where p(k) denotes the number of non-increasing partitions of integer k

1-hypergroups of small sizes

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 \u3b2. Many of these hypergroups can be obtained using the aforesaid hypergroup construction

The LV-hyperstructures

The largest class of hyperstructures is the one which satisfy the weak properties and they are called H v -structures introduced in 1990. The H v(c)-structures have a partial order (poset) on which gradations can be defined. We introduce the LV-construction based on the Levels Variable

Isomorphism classes of the hypergroups of type U on the right of size five

AbstractBy means of a blend of theoretical arguments and computer algebra techniques, we prove that the number of isomorphism classes of hypergroups of type U on the right of order five, having a scalar (bilateral) identity, is 14751. In this way, we complete the classification of hypergroups of type U on the right of order five, started in our preceding papers [M. De Salvo, D. Freni, G. Lo Faro, A new family of hypergroups and hypergroups of type U on the right of size five, Far East J. Math. Sci. 26(2) (2007) 393–418; M. De Salvo, D. Freni, G. Lo Faro, A new family of hypergroups and hypergroups of type U on the right of size five Part two, Mathematicki Vesnik 60 (2008) 23–45; M. De Salvo, D. Freni, G. Lo Faro, On the hypergroups of type U on the right of size five, with scalar identity (submitted for publication)]. In particular, we obtain that the number of isomorphism classes of such hypergroups is 14865

EL-hyperstructures: an overview

This paper gives a current overview of theoretical background of a special class of hyperstructures constructed from quasi / partially or dered (semi) groups using a construction known as the "Ends lemma". The paper is a collection of both older and new results presented at AHA 2011