connecting properties on m-sets

The notion of the action of a group (or monoid in a general way) on a set is a fundamental one, perhaps even more so than that of a group (monoid) itself: groups (monoids) derive their interest from their actions. here, for more studing one this structure, we first introduce and describe the notion of connected m-sets, in which m is a monoid. then we introduce the strongly connected m-set and give the characterization of this notion and investigate the relation between these kinds of connectedness. in the meanwhile, we in particular give the additional properties that these kinds of connected finitely supported m-sets have.