Abstract – Via the transposition of the theorems of Birkhoff, ChaunierLygeros and LygerosMizony on Groups to PHypergroups, we explicit some relations between posets and hypergroups.
1. Definitions
is a hypergroup if : H*H→p(H) is an associative hyperoperation for which the reproduction axiom hH=Hh=H is valid for every h of H [5].
A hypergroup (H,.) is called cyclic with finite period with respect an element h of H if there exists an integer v such that H=h^{1}∪h^{2}∪...∪h^{v}[3].
Let (G,.) be a group and P any nonempty subset of G. Then the Phyperoperation P* is defined as follows: P*: G*G→p(G) : (x,y)→xP*y=xPy. If the reproduction axiom is valid then the hyperstructure is a Phypergroup [6].
2. Theorems
Theorem Birkhoff [1]: If G is group then there exists a poset which automorphisms group is isomorphic to G.
Theorem Birkhoff [1]: If G is a finite group of cardinal a then there exists a poset which automorphisms group is isomorphic to G and with cardinal is equal to a^{2}+a.
Theorem ChaunierLygeros [2]: Let a a prime number and n the minimal number of vertices of posets which have a automorphisms group of cardinality a. Then:
(i) n=a if a=2;
(ii) n=3a if a=3,5 or 7;
(iii) n=2a if a>7.
Posets which realize the minimum and have a minimal number of relations are respectively:
(i) ({x_{0},x_{1}}) with x_{0} and x_{1} incomparable;
(ii) ({x_{0},…,x_{a1},y_{0},…y_{a1},z_{0},…,z_{a1}}) with x_{i}< y_{i}< z_{i} and x_{i}< z_{j} if ji≡1 (mod a);
(iii) ({x_{0},…,x_{a1},y_{0},…y_{a1}}) with x_{i}< y_{j} if ji≡0,1 or 3 (mod a).
Theorem LygerosMizony [5]: If G is a finite group of cardinal a, non direct product of two groups, generated by elements which are two by two of distinct order then there exists a poset which automorphisms group is isomorphic to G and with cardinal is equal to 3a.
3. New Theorems
Theorem 3.1: A poset P_{o} can be associated to every Phypergroup.
Proof: By the definition of a Phypergroup we have: G←→< G,P* > Phypergroup
and with the theorem of Birkhoff [1] we can associate a poset P_{o} to a group G via its automorphisms group Aut(P_{o}). So we have: P_{o} ←→ Aut(P_{o}) ≈ G←→ < G,P* > Q.E.D.
Theorem 3.2: A poset P_{o} of cardinality a^{2}+a can be associated to every Phypergroup with G of cardinality a.
Proof: Corollary of the second theorem of Birkhoff. Q.E.D.
Theorem 3.3: Let a a prime number, Phypergroup and G a group of cardinality a=2 or 3,5,7 or 11 and more then there is an associated poset P_{o} of cardinality respectively a or 3a or 2a which has an automorphism group of order a.
Proof: Corollary of the theorem of ChaunierLygeros.
Q.E.D.
Theorem 3.4: Let Phypergroup and G a finite group of cardinal a, non direct product of two groups, generated by elements which are two by two of distinct order then there exists a poset which automorphisms group is isomorphic to G and with cardinal is equal to 3a.
Proof: Corollary of the theorem of LygerosMizony.
Q.E.D.
4. Questions
Question 1: Can we associate a general hypergroup to a poset?
Question 2: Can we associate a poset to a general hypergroup?
References
[1] Birkhoff G., 1946. Sobre los grupos de automorphimos. Revista Union Math. Arg., 11. pp.155157.
[2] Chaunier C. and N. Lygeros 1994. Posets minimaux ayant un groupe d’automorphismes d’ordre premier. C.R.Acad.Sci. Paris, t.318, Série I, p. 695698.
[3] De Salvo M. and Freni D. 1981. Semiipergruppi ciclici. Atti Sem. Mat. Fis. Un. Modena, 30, pp.4459.
[4] Lygeros N. and Mizony M.1996. Construction de posets dont le groupe d’automorphismes est isomorphe à un groupe donné. C.R.Acad.Sci. Paris, t.322, Série I, pp. 203206.
[5] Marty F. 1934. Sur une généralisation de la notion de groupe. 8th Congress Math. Scandinaves, Stockholm, pp.4549.
[6] Vougiouklis Th. 1990. Isomorphisms on Phypergroups and cyclicity. Ars Combinatoria 29A , pp.241245.
