0-adequate semigroups

We introduce the concept of 0-adequate semigroups.It is a special inverse semigroup.We give the characterizations of 0-adequate semigroups.We discuss the largest congruence μL contained in ■0 on a right 0-adequate semigroup with semilattice of idempotents E and the largest congruence μ contained in ■0 on a 0-adequate semigroup with semilattice of idempotents E.We prove that if S is a right 0-A type semigroup with semilattice of idempotents E,then S/μL≌E if and only if S is a strong semilattice of S0 left inverse left cancellative monoids.Moreover,we show that if S is a 0-adequate semigroup with semilattice idempotens,then S/μ≌E if and only if S is a strong semilattice of S0 inverse cancellative monoids.