On Hypergroups with a xcexb2Class of Finite Height In every hypergroup, the equivalence classes modulo the fundamental relation    xcexb2    are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a    xcexb2   class and we analyze properties of hypergroups where the height of a    xcexb2   class coincides with its cardinality. As a consequence, we obtain a new characterization of 1hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one    xcexb2   class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is     n xe2x89xa4 4    , apart from isomorphisms.

Dualization in lattices given by implicational bases Abstract   It was recently proved that the dualization in lattices given by implicational bases is impossible in outputpolynomial time unless  P  NP . In this paper, we show that this result holds even when the premises in the implicational base are of size at most two. Then we show using hypergraph dualization that the problem can be solved in output quasipolynomial time whenever the implicational base has bounded independentwidth, defined as the size of a maximum set of implications having independent conclusions. Lattices that share this property include distributive lattices coded by the ideals of an interval order, when both the independentwidth and the size of the premises equal one.