Hypergroupoids on Partially Ordered Sets

Hypergroupoids on Partially Ordered Sets
Josef Zapletal
European Polytechnical Institute, LLC.
686 04 Kunovice, Czech Republic,
e-mail: [email protected]
A hypergroupoid (or a multigroupoid) is a pair ( M, ◦) where M is a nonempty set
and ◦ : M × M → P ∗ (M ) is a binary hyperoperation also called a multioperation.
P ∗ (M ) is the system of all nonepmty subsets of M . Partially ordered set M with the
ordering ≤ with the greatest element I is in this article denoted with M = (M, ≤, I). On
M = (M, ≤, I) for arbitrary x, y ∈ M , we define a binary hyperoperation ◦ as follows:
x ◦ y = {min(X ∩ Y )}.
where X = {xi | xi ≥ x} and Y = {yi | yi ≥ y} We then denote the set M with the
defined binary operation with M = ( M ≤, ◦, I). It is proved that the hyperoperation
◦ on (M = ( M ≤, ◦, I) is idempotent and commutative but not associative. Hence the
partially ordered set M with the operation ◦ is a commutative hypergroupoid.
In the beginning of the second chapter a definition of congruence on a commutative
hypergroupoid M is given. By a congruence we call a relation of equivalence ρ on M
such that for every quadruple of elements a1 , a2 , b1 , b2 ∈ M for which a1 ρ b1 , a2 ρ b2 the
following holds: For every x ∈ a1 ◦ a2 there exists y ∈ b1 ◦ b2 and for every y 0 ∈ b1 ◦ b2 there
exists x0 ∈ a1 ◦ a2 with the property xρ y and x0 ρ y 0 . See [4] p.151 and [10]. It is shown
that the relation of substitutabality Ξ(M,L) satisfies this definition. The distinguishing
subsets of commutative hypergroupoids are studied in the third chapter. The fourth
chapter contains a concrete partially ordered set Q (Figure 1) with the hyperoperation ◦.
The congruence Ξ(Q,L) with its partition of the set Q is given as an example. The other
example demonstrates a distinguishing subset of Q.
wI
Partially ordered set Q
@
@
@
@
@w l
wj
wk @
@
@
@
@
@
@
w g @w h @w i
@
@
@
@
@
@
w d @w e @w f
@
@
@
@
@
@
w a @w b @w c
@
@
@
@w 0
Figure 1
References
[1] J. Chvalina. General Algebra and Ordered Sets. Proceedings of the Summer School
1994. Hornı́ Lipová, Czech Repuplic, September 4 - 12, 1994. Department of Algebra
and Geometry Palacký University Olomouc, Olomouc, Czech Republic.
[2] J. Chvalina. Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups.
Masarykova Univerzita, Brno, 1995 (In Czech)
[3] J. Chvalina. From Functions of One Real Variable to Multiautomata. 2. Žilinská
didaktická konferencia, did ZA 2005, 1/4
[4] J. Chvalina. Funkcionálnı́ grafy, kvaziuspořádané množiny a komutativnı́ hypergrupy.
Masarykova Univerzita, Brno, 1995
[5] J. Chvalina. Š. Hošková. Abelization of quasi-hypergroups as reflextion. Second Conf.
Math. and Physics at Technical Universities, Military Academy Brno, Proceedings of
Contributions, MA Brno (2001), 47-53 (In Czech).
[6] J. Chvalina, L. Chvalinová. Multistructures determined by differential rings. Arch.
Mat., Brno (2000), T.36, CDDE 2001 issue, 429 -434.
[7] P. Corsiny. Prolegomena of Hypergroup Theory ,Aviani Editore, Tricestimo, 1993.
[8] P. Corsiny. Hyperstructures associated with ordered sets.Proc. of the Fourth Panhellenic Conference on Algebra and Number Theory, in printing onBull. of the Greek
(Hellenic) Mathematical Society.
[9] D., A. Hort. A construction of hypergroups from ordered structures and their morphisms. Proceedings of Algebraic Hyperstructures and Applications, Taormina, 1999,
J. of Discrete Math.
[10] J. Karásek. On general algebras, Arch. Mat. Brno 2 (1966) 157 - 175.
[11] E. S. Ljapin. Polugruppy, Moskva 1960, (in Russian).
[12] R. Migliprato, G. Gentile. Feebly associative hypergroupoids. Proceedings of the International Conference on Finite Geometries, Perugia, Italy 1992, (1993), 259-268.
[13] I. G. Rosenberg. Hypergroups and join spaces determined by relations. Italian Journal
of Pure and Applied Mathematics, no 4(1998), 93-101.
[14] G. Szász. Introduction to Lattice Theory. Akadémia Kiadó, Budapest 1963.
[15] J. Zapletal. Distinguishing Subsets of Semigroups and Groups, Arch. Math. Brno,
1968, Tomus 4, Fasc. 4,241-250.