24 research outputs found
Directed Graphs representing isomorphism classes of C-Hypergroupoids
We investigate the relation of directed graphs and hyperstructures by virtue of the graph hyperoperation. A new class of graphs arises in this way representing isomorphism classes of C-hypergroupoids and we present the 17 such graphs that correspond to the 73 C-hypergroupoids associated with binary relations on three element sets. As it is shown they constitute an upper semilattice with respect tograph inclusion
Enumeration of Rosenberg-type hypercompositional structures defined by binary relations
AbstractEvery binary relation Ď 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 Rosenbergâs hyperoperation are characterized, constructed and enumerated using symbolic manipulation packages. Moreover, the hyperoperation given by xâx={zâH|(z,x)âĎ} and xây=xâxâŞyây is introduced and connected to Rosenbergâs hyperoperation, which assigns to every (x,y) the set of all z such that either (x,z)âĎ or (y,z)âĎ
History and new possible research directions of hyperstructures
We present a summary of the origins and current developments of the theory of algebraic hyperstructures. We also sketch some possible lines of research
An Overview of Topological and Fuzzy Topological Hypergroupoids
On a hypergroup, one can define a topology such that the hyperoperation is pseudocontinuous or continuous.This concepts can be extend to the fuzzy case and a connection between the classical and the fuzzy (pseudo)continuous hyperoperations can be given.This paper, that is his an overview of results received by S. Hoskova-Mayerova with coauthors I. Cristea , M. Tahere and B. Davaz, gives examples of topological hypergroupoids and show that there is no relation (in general) between pseudotopological and strongly pseudotopological hypergroupoids. In particular, it shows a topological hypergroupoid that does not depend on the pseudocontinuity nor on strongly pseudocontinuity of the hyperoperation
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
Neutrosophic Hypercompositional Structures defined by Binary Relations
The objective of this paper is to study neutrosophic hypercompositional structures arising from the hypercompositions derived from the binary relations on a neutrosophic set. We give the characterizations of hypergroupoids,quasihypergroups, semihypergroups, neutrosophic hypergroupoids, neutrosophic quasihypergroups, neutrosophic semihypergroups and neutrosophic hypergroups
C-hypergroupoids obtained by special binary relations
AbstractIn this paper we deal with the partial or non-partial C-hypergroupoids which are associated with special binary relations defined on H, such as Reflexive, Symmetric, Cyclic and Transitive. Basic properties are investigated and various characterizations are given. The main tool to study the previous special classes of hypergroupoids is the fundamental relation βâ (i.e. the smallest equivalence relation such that the quotient of a hypergroupoid (partial or not) is a groupoid (partial or not)
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
Atomistic subsemirings of the lattice of subspaces of an algebra
Let A be an associative algebra with identity over a field k. An atomistic
subsemiring R of the lattice of subspaces of A, endowed with the natural
product, is a subsemiring which is a closed atomistic sublattice. When R has no
zero divisors, the set of atoms of R is endowed with a multivalued product. We
introduce an equivalence relation on the set of atoms such that the quotient
set with the induced product is a monoid, called the condensation monoid. Under
suitable hypotheses on R, we show that this monoid is a group and the class of
k1_A is the set of atoms of a subalgebra of A called the focal subalgebra. This
construction can be iterated to obtain higher condensation groups and focal
subalgebras. We apply these results to G-algebras for G a group; in particular,
we use them to define new invariants for finite-dimensional irreducible
projective representations.Comment: 14 page