About some separation axioms in semitopological inverse semigroups (Q1277628)

A semitopological semigroup \(S\) is said to be inverse if its idempotents are commutative under multiplication and for all \(x\in S\), there exists \(y\in S\) such that \(xyx=x\) and \(yxy=y\). The inverse semigroup is said to be maximized if for every \(x\in S\) there exists \(m\in M_S=\{m\in S\mid\forall y\in S\), \(mm^{-1}y=y\) and \(m^{-1}my=y\}\) such that \(x\leq m\). In this paper, the author mainly studies the semitopological maximized inverse semigroups, in particular, the separation axioms between \(T_1\) and \(T_2\), \(T_3\) and also those which satisfy certain order conditions. The results obtained by A. Conte in 1969 and 1971 are hence enriched. Although, the axioms \(T_0\) and \(T_2\) are equivalent on topological groups, this equivalence will disappear if the connections between the topological and the algebraic structures are weakened.