Enumeration of Varlet and Comer Hypergroups

In this paper, we study hypergroups determined by lattices introduced by Varlet and Comer, especially we enumerate Varlet and Comer hypergroups of orders less than 50 and 13, respectively. 1 Basic definitions and results An algebraic hyperstructure is a natural generalization of a classical algebraic structure. More precisely, an algebraic hyperstructure is a non-empty set H endowed with one or more hyperoperations that associate with two elements of H not an element, as in a classical structure, but a subset of H . One of the interests of the researchers in the field of hyperstructures is to construct new hyperoperations using graphs [18], binary relations [2, 5, 7, 8, 9, 11, 15, 21, 23], n-ary relations [10], lattices [16], classical structures [13], tolerance space [12] and so on. Connections between lattices and hypergroupoids have been considered since at least three decades, starting with [24] and followed by [3, 14, 17]. This paper deals with hypergroups derived from lattices, in particular we study some properties of the hypergroups defined by J.C. Varlet [24] and S. Comer [3] that called here Varlet hypergroups and Comer hypergroups, respectively. Using the results of [1, 22] the electronic journal of combinatorics 18 (2011), #P131 1 we enumerate the number of non isomorphic Varlet and Comer hypergroups of orders less than 50 and 13, respectively. Let us briefly recall some basic notions and results about hypergroups; for a comprehensive overview of this subject, the reader is referred to [4, 6, 25]. For a non-empty set H , we denote by P(H) the set of all non-empty subsets of H . A non-empty set H , endowed with a mapping, called hyperoperation, ◦ : H −→ P(H) is named hypergroupoid. A hypergroupoid which satisfies the following conditions: (1) (x ◦ y) ◦ z = x ◦ (y ◦ z), for all x, y, z ∈ H (the associativity), (2) x ◦ H = H = H ◦ x, for all x ∈ H (the reproduction axiom) is called a hypergroup. In particular, an associative hypergroupoid is called a semihypergroup and a hypergroupoid that satisfies the reproduction axiom is called a quasihypergroup. If A and B are non-empty subsets of H , then A ◦ B = ⋃ a∈A,b∈B a ◦ b. Let (H, ◦) and (H , ◦) be two hypergroups. A function f : H −→ H ′ is called a homomorphism if it satisfies the condition: for any x, y ∈ H , f(x ◦ y) ⊆ f(x) ◦ f(y). f is a good homomorphism if, for any x, y ∈ H , f(x ◦ y) = f(x) ◦ f(y). We say that the two hypergroups are isomorphic if there is a good homomorphism between them which is also a bijection. Join spaces were introduced by W. Prenowitz and then applied by him and J. Jantosciak both in Euclidian and in non Euclidian geometry [19, 20]. Using this notion, several branches of non Euclidian geometry were rebuilt: descriptive geometry, projective geometry and spherical geometry. Then, several important examples of join spaces have been constructed in connection with binary relations, graphs and lattices. In order to define a join space, we need the following notation: If a, b are elements of a hypergroupoid (H, ◦), then we denote a/b = {x ∈ H | a ∈ x ◦ b}. Moreover, by A/B we intend the set