Slowly oscillating functions and closed left ideals of \(\beta \)S (Q1005158)

For a discrete semigroup \(S\), the Stone-Čech compactification \(\beta S\) has a natural structure of compact right topological semigroup. Let \(\varphi\) be a filter on \(S\). A function \(f:S\rightarrow [0,1]\) is slowly oscillating in the direction of \(\varphi\) if for every \(\varepsilon>0\) and for every finite subset \(F\) of \(S\), there exists \(A\in\varphi\) such that \(diam(Ft\cup \{t\})<\varepsilon\) for every \(t\in A\). Let \(S\) be an infinite discrete semigroup with a point \(s\) such that \(st\neq t\) for any \(t\in S\). The authors prove that every closed left ideal in \(\beta S\) with countable base is determined by a function on \(S\) which is slowly oscillating in the direction of some filter. For an infinite discrete group, the same result was proved in [\textit{M. Filali} and \textit{I. Protasov}, Topology Appl. 154, No.~3, 561--566 (2007; Zbl 1180.54043)].