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A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8).There is a well-known internal characterisation of right PP monoids using the relation X* which is defined as follows. On a semigroup S, (a,b)E.Z£* if and only if the elements a,b of S are related by Green's relation 2? in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each i£*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each 2?*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation 91* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate will be called an adequate semigroup.Any inverse semigroup is adequate and in investigating adequate semigroups it is natural to look for analogues of results for inverse semigroups. In (6), Howie determines the greatest idempotent separating congruence n on an inverse semigroup S and shows that Slfi is isomorphic to the semilattice of idempotents of S if and only if the idempotents in S are central. In an adequate semigroup there need not be a greatest idempotent separating congruence. However, on an inverse semigroup, /x is the largest congruence contained in 7R,. Defining Stf* to be X* D91*, we determine, in Section 2, the largest congruence ix L contained in X* on a right adequate semigroup and hence find the largest congruence fi contained in "3t* on an adequate semigroup. We also find that the analogue of the second result of Howie mentioned above holds.It is well known (see (10) or (7, Chapter V)) that if S is an inverse semigroup with semilattice of idempotents E, then there is an idempotent separating homomorphism 0 from S into the Munn semigroup T E such that 0 °

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SynopsisA general theory for a class of abundant semigroups is developed. For a semigroup S in this class let E be its set of idempotents and <E> the subsemigroup of S generated by E. When <E> is regular there is a homomorphism with a number of desirable properties from S onto a full subsemigroup of the Hall semigroup T<E>. From this fact, analogues of results in the regular case are obtained for *-simple and ℐ*-simple abundant semigroups.

We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y*T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X* acts doubly on the semilattice [Formula: see text] of idempotents of FIM(X) and that FAM(X) is embedded in [Formula: see text]. Finally we show that every weakly E-ample monoid has a proper ample cover.