Noble inverse semigroups with bisimple core (Q579423)

An inverse semigroup is said to be noble if and only if it possesses a faithful transitive representation by one-to-one partial transformations on a set. An old problem - how to characterize noble inverse semigroups - is solved in the present paper for inverse semigroups with a bisimple (0- bisimple) ideal. A subset B of an inverse semigroup S is called a basis if each element of S is the least upper bound (with respect to the natural order on S) of a subset of B. The main result: an inverse semigroup S with a nonzero bisimple (0-bisimple) ideal B is noble if and only if B is a basis of S. The author also writes that a description of noble inverse semigroups in the general case will be considered in a separate publication.