Envelopes of syndetic subsemigroups of the acting topological semigroup in a semiflow

A theory of semiflows with a discrete acting topological semigroup was developed in the 2000 paper by D. Ellis, R. Ellis and M. Nerurkar (2000) [3]. A theory for the case of an arbitrary acting topological semigroup has still to be developed. This paper can be considered as the beginning of an attempt in that direction.We introduce the notion of the envelope of a g-syndetic subsemigroup of the acting topological semigroup in a semiflow and prove various properties of this notion. We give some examples and deduce some statements from other papers as corollaries of our statements.