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...

In this paper the concept of anti fuzzy ideals of a Γsemigroup has been introduced and some important characterizations have been obtained. Among other results a regular Γ-semigroup has been characterized in terms of anti fuzzy ideals. Finally we show that the set of all the anti fuzzy ideals of a Γ-semigroup forms a semilattice. 2010 AMS Classification: 20M99, 08A72, 20M12

2-We have a group G or a semigroup S with a set of (semigroup) generators A. We have a natural mapping ! : A* " G. Various notions of classes of groups and semigroups give rise to (the possibility of) effective computation. In the cases we will consider, we have L # A + = A*-{$} such that L is regular and L! = G.

A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete nhomomorphism but not a V-homomorphism. However, the trace classes are intervals in the complete lattice of right congruences. In contrast, each kernel class has a maximum element, namely the principal right congruence on the kernel, but in general there is no m...