the semigroup of ultrafilters near zero as the adequacy kernel of a partial semigroup

‎let $t$ be a dense subsemigroup of $((0,infty),+)$ and put $s = t cap (0,1)$‎. ‎in this paper we show that $(s,dot{+})$ is a commutative adequate partial ‎semigroup‎, ‎where $x dot{+} y = x+y$ whenever $x+y in s$‎. ‎first‎, ‎we prove that the semigroup of ultrafilters near zero of $s$ is an‎ ‎adequacy kernel of $s$; consequently‎, ‎every result that holds for ‎commutative adequate partial semigroups also holds for the semigroup of‎ ‎ultrafilters near zero‎. ‎second‎, ‎we demonstrate that $(s,dot{+})$ does not satisfy the $(sfc)$‎ ‎condition for commutative adequate partial semigroups‎. ‎finally‎, ‎we propose a f{o}lner density equipped with a suitable operation on $s$‎, ‎which aids the study of density near zero‎.