A SIMPLIFICATION OF D ’ ALARCAO ’ S IDEMPOTENT SEPARATING EXTENSIONS OF INVERSE SEMIGROUP

In [2] D'Alarcao states necessary and sufficient conditions for the attainment of an idempotent-separating extension of an inverse semigroup. To do this D'Alarcao needed essentially three mappings satisfying thirteen conditions. In this paper we show that one can achieve the same results with two mappings satisfying eight conditions.

-I (i) af (a a)f for all a A; -I -I (i i) a ga ga a for all a A and g E(A); and (i i) ga ag for all a A and g E(A).
PROOF. (i) Since af E(B), by Lallement's result [3], there exists -I -I (iii) By part (ii), ga a(a ga) a(ga a) ag.This completes the proof of the emma.
Let A and B be inverse semigroups and let f be a homomorphism of A onto E(B) which is one-to-one on E(A). (The existence of such an f implies that A is a semi lattice of groups; see e.g.[4].)Suppose there Moreover, let y'S*-B be defined by (b,a)y b.
THEOREM I.Under the above conditions, (S*,y) is an idempotentoseparating extension of A by B.Moreover, YIA f.
PROOF.Note that since fof-I is idempotent separating, fof-I .
Our proof that S* is regular is the same as that found in [2 but is more informative.Let  pf.
Since a, p E(A), a p. Consequently y is one-to-one on E(A*).It is obvious that Y IA ,= f" Since A* A, this completes the proof of the theorem.
Since our two mappings and eight properties are the same as, but fewer than, those used by D'Alarcao, the following theorem has been proven by D'Alarcao ([2], Theorem 2).
THEOREM 2. Let A and B be inverse semi groups and let (S,f) be an and there exists a mapping w of B x B into A satisfying (PI) through (P8).Moreover, S is isomorphic to S*.
a)f.Since f is one-to--I -I -I one on E(A), we have a ga (ga) ga ga a.
is a mapping w of B x B into A (denoted by (b,s A and b, c B. (PS) For each b, c s E(B), b c eg where {e} E(A) #I bf -I and {g} E(A) F cf -I.-I LeT S* {(b,a)-b B and a (b b)f}.Let equality in S* be defined in the usual manner and let multiplication be defined by (b,a)(c,p) (bc,bCacp).

E
(b,a) e S* and let x ea)= (b,a) since {e} E(A) i' (b-lb)f -I Consequently, S* is regular As shown in [2, it follows from (P5) and (P6) that if (g,e) E(S*) then g E(B) and {e} E(A) gf-I.Consider the subset A* {(b,a)-b e E(B)} c_ S* which contains E(S*).Let 6"A* / A by (b,a) + a.To show 6 is a homomorphism, let (b,a), (c p)A* Then a ea and p gp where {e} which obviously has A for its range.It fol lows from the definitions of S* and 8 that is one-to-one.Consequently the dempotents of S* commute -I It is obvious that y is a homomorphism of S* onto B and {E(B)}y A*.If (b,a), (c,p) E(A*) such that (b,a)y (c,p)y, then b c and af b c idempotentoseparating extension of A by B. Then for each b B there exists a mapping U b of A into A (denoted by aU b b a