We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I n λ of the rank 6 n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. W...

The symmetric inverse semigroup I(X) on a set X is the collection of all partial bijections between subsets with composition as algebraic operation. We study minimal Hausdorff topology I(X). present some characterizations it. When countable such Polish.

For strongly continuous semigroups on a Hilbert space, we present a short proof of the fact that the left-inverse of a left-invertible semigroup can be chosen to be a semigroup as well. Furthermore, we show that this semigroup need not to be unique.