No, the ids are unique for each element.
I changed the grid code myself and added the prefix ts.p.pager to all pager elements like _first, _prev, _next and _last so the ids of these elements changed from “_next” to “MyGridPagerId_next”. Now the grid does what i expected.
Take a look at the demo page, the ids of the elements in the pager are not unique – even if this will work for you it is not correct in the dom.
Cheers J