  This paper constructs a foundation to analyze semi-group actions, group
actions, filtrations, and decompositions in a unified manner. In fact, though
the studies of decomposition can be applied to foliated spaces and group
actions, they can not be applied to semi-group actions and filtrations in
general because filtration and the set of orbits of a semi-group need not be
decompositions of the base spaces. To analyze these concepts in a unified
manner, we introduce a concept of a semi-decomposition which is a natural
generalization of these concepts because similar relations among recurrence and
their variants to group actions and decompositions hold for
semi-decompositions. On the other hand, we demonstrate the difference between
the recurrent concepts for group actions and those even for semi-group actions.
