in this thesis, first the notion of weak mutual associativity (w.m.a.) and the necessary and sufficient condition for a $(l,gamma)$-associated hypersemigroup $(h, ast)$ derived from some family of $lesssim$-preordered semigroups to be a hypergroup, are given. second, by proving the fact that the concrete categories, semihypergroups and hypergroups have not free objects we will introduce the notion of weak free semihypergroup for the classes of semihypergroups ($mathbf{shypgrp}$) and extension complete semihypergroups ($mathbf{ecs}$). also we give a necessary and sufficient condition for being weak free semihypergroups in the category $mathbf{ecs}$. the existence of proper weak free object for the class of $mathbf{ecs}$ has been proved. third, we introduce a strongly regular equivalence relation $m*$, on a hypergroup $h$ such that in a special case the quotient $frac{h}{m*}$ is a cyclic group. we also investigate the transitivity condition for $m*$ and a characterization of the new derived hypergroup $d_c(h)$. then we define the notion of hyperaction of a hypergroup on a nonempty set and also the notion of index of a subhypergroup in a hypergroup, as a generalization of the concept of action of a group on a nonempty set and the notion of index of a subgroup in a group, respectively. some properties such as the generalized orbit-stabilizer theorem, are investigated. in particular, we introduce a construction of a hypergroup from a hyperaction. at the last chapter of this thesis we will assign a generalized state hypergroup to a nondeterministic automata which associates from a hyperaction.