A NOTE ON STRONGLY E-REFLEXrVE INVERSE SEMIGROUPS

In contrast to the semilattice of groups case, an inverse semigroup S which is the union of strongly ^-reflexive inverse subsemigroups need not be strongly £-reflexive. If, however, the union is saturated with respect to the Green's relation <3), and in particular if the union is a disjoint one, then 5 is indeed strongly £-reflexive. This is established by showing that fy -saturated inverse subsemigroups have certain pleasant properties. Finally, in contrast to the £-unitary case, it is shown that the class of strongly £-reflexive inverse semigroups is not closed under free inverse products. The reader is referred to [1], [2] for the basic theory of inverse semigroups, including the theory of free inverse products. Recall from [4], [5] that an inverse semigroup S is said to be strongly E-reflexive whenever S is a semilattice of E-unitary inverse semigroups, or alternatively, whenever there exists a semilattice of groups congruence 17 on S such that only idempotents are linked to idempotents under rj. In [4], [5] we studied this class of semigroups and showed that many of the properties of semilattices of groups and of E-unitary inverse semigroups generalise to this class, albeit sometimes in a weaker form. We continue this line of investigation here. In what is by now a classic theorem, Clifford showed that an inverse semigroup which is a union of groups is a semilattice of groups. We ask to what extent this is true for strongly .E-reflexive inverse semigroups. It is already known that a semilattice of strongly E-reflexive inverse semigroups is again strongly E-reflexive [5]. The following simple example shows that we cannot hope for a full generalisation of Clifford's theorem. Consider the bisimple inverse to-semigroup S(G, a), where the endomorphism a of the group G is not injective. As noted in [4, p. 341], S(G, a) is not strongly E-reflexive. However, using [1, Lemma 1.31], it is easily seen that S(G, a) is a union of its maximal subgroups and copies of the bicyclic semigroup, and these are all E-unitary. The restriction we require will now be given, and the example just noted would seem to indicate that it is the weakest possible. Let S be an inverse semigroup with semilattice of idempotents E. Let U be an inverse subsemigroup of S which is tf)-saturated in the sense that x ^ y E U implies x E U, where tf) denotes the usual Green's relation on S. The maximal group homomorphic image of U is denoted by U with ¿7 denoting the image of u (u E U). Let U' = [x E S\x > u for some u E U); note that U' may equal S. Received by the editors March 23, 1979. AMS (MOS) subject classifications (1970). Primary 20M10. © 1980 American Mathematical Society 0002-9939/80/0000-0301/$01.75 352 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use STRONGLY ¿'-REFLEXIVE INVERSE SEMIGROUPS 353 The first result shows that U' has some pleasant properties. Proposition, (i) U' is an inverse subsemigroup of S which contains U, and xy £ U' implies x £ V and y £ U'. (ii) The rule: x<¡>=üifx>u£U and x = 0 otherwise, gives a well-defined homomorphism : S —» U° such that u £ U => xx ~ ' > xyy ~xx~x > uu~x => x > uu~xx '51 « =* x E U', since U is <$ -saturated and <3l C tf). Dually, y £ U'. The remainder of the result is easily proven. (ii) Suppose x £ U' with x > u £ U and x > v £ U. Then u = ex, v = fx where e = uu~x £ U n E,f = w~x £ U n E. Hence efu = efv, and ef £ E n U, so that U = v. It is then almost immediate that : S —> T with s having /th component $(/>„ ; £ I. Now S is a semilattice of groups, since T is. Suppose that x g,f > g, and e,f are incomparable. Let S be the semilattice of groups Ge u G} u Gg where Ge, Gg are trivial and Gj is the cyclic 2-group; let T be the semilattice of groups He u Hf u Hg License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use