  Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise
syndetic set contains k term arithmetic progressions and such collection is
also piecewise syndetic in Z: They used algebraic structure of beta N. The
above result was extended for arbitrary semigroups by Bergelson and Hindman,
again using the structure of Stone-Cech compactification of general semigroup.
Beiglboeck provided an elementary proof of the above result and asked whether
the combinatorial argument in his proof can be enhanced in a way which makes it
applicable to a more abstract setting. In a recent work the second author of
this paper and S.Jana provided an affirmative answer to Beiglboeck's question
for countable commutative semigroup. In this work we will extend the result of
Beiglboeck in different type of settings.
