On ideals of a semilattice
We present some special properties of inverse categories with split idempotents. First, we examine a Clifford-Leech type theorem relative to such inverse categories. The connection with right cancellative categories with pushouts is illustrated by simple examples. Finally, some basic properties of inverse categories with split idempotents and kernels are studied in terms of split idempotents which generate (right or left) principal ideals of annihilators.
In this paper the equivalence on a semigroup in terms of a set of idempotents in is defined. A semigroup is called a -liberal semigroup with as the set of projections and denoted by if every -class in it contains an element in . A class of -liberal semigroups is characterized and some special cases are considered.