We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These rep...

In contrast to the semilattice of groups case, an inverse semigroup S which is the union of strongly ^-reflexive inverse subsemigroups need not be strongly £-reflexive. If, however, the union is saturated with respect to the Green's relation <3), and in particular if the union is a disjoint one, then 5 is indeed strongly £-reflexive. This is established by showing that fy -saturated inverse sub...

The symmetric inverse monoid In is the set of all partial permutations of an n-element set. The largest possible size of a 2-generated subsemigroup of In is determined. Examples of semigroups with these sizes are given. Consequently, if M(n) denotes this maximum, it is shown that M(n)/|In| → 1 as n →∞. Furthermore, we may deduce, the already known fact, that In embeds as a local submonoid of an...

Left restriction semigroups have appeared at the convergence of several flows of research, including the theories of abstract semigroups, of partial mappings, of closure operations and even in logic. For instance, they model unary semigroups of partial mappings on a set, where the unary operation takes a map to the identity map on its domain. This perspective leads naturally to dual and two-sid...