On Some Embedding Theorems for Inverse Semigroups

A semilattice decomposition of an inverse semigroup has good internal mapping properties. These are used to give natural proofs of some embedding theorems, which were originally proved in a rather artificial way. The reader is referred to [1] for the basic theory of inverse semigroups. In an earlier paper [3] we proved the following embedding result: (1) An E-unitary inverse semigroup is isomorphic to an inverse subsemigroup of a semidirect product of a semilattice and a group. The class of over-semigroups mentioned in (1) will be denoted by 6,. The proof of (1) used McAlister's P-theorem [2]. Later, in [4], we generalised the TMheory to arbitrary inverse semigroups and proved a general embedding theorem. This result was then used in [6] to prove the following generalisation of (1): (2) A strongly 7s-reflexive inverse semigroup (i.e. one which is a semilattice of Ti-unitary inverse semigroups) is embedded in a strong semilattice of inverse semigroups each of which is a member of <2j. The class of over-semigroups mentioned in (2) will be denoted by C2. Finally, in [7], (2) was used to prove the following result: (3) An inverse semigroup is strongly Ts-reflexive (if and) only if it is a subdirect product of Ti-unitary inverse semigroups with zero added possibly. Now a member of C,, with zero added possibly, is a member of ß2 in a rather trivial way. Moreover, it is relatively easy to see that C2 is closed under arbitrary direct products. From this it follows that not only can (3) be deduced from (2), but that (2) can be deduced from a combination of (1) and (3). The proof of (1) in [3], which is based on the original 'external' or 'abstract' TMheorem [2], involves the intuitively plausible idea of suitably completing the semilattice component of a TMriple so that the group component can act on it without constraint (see [3] for details). Meanwhile, Schein [9] had given a succinct and conceptually clear proof of the TMheorem, which, as we shall see, gives (1) as an almost immediate byproduct. In [5] we showed that Schein's 'internal' approach could be successfully adapted to give the generalized P-theory and to prove the generalised embedding theorem. In contrast to this near ideal state of affairs for the Ti-unitary case, the general embedding theorem (and so its consequences (2) and (3)), whether in 'external' or Received by the editors September 29, 1980. 1980 Mathematics Subject Classification. Primary 20M10. © 1981 American Mathematical Society 0002-9939/81/0000-0401/$01.75