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If S is a regular semigroup with set of idempotents ES t h e n a n i n verse subsemigroup of S is called an inverse transversal of S if S contains a unique inverse x of each element x of S. The class of regular semigroups with inverse transversals was introduced by Blyth and McFadden 1 in 1982. This large class contains, for example, elementary rectangular bands of inverse semigroups 10 , naturally ordered regular semigroups with a biggest idempotent 7 , regular 4-spiral semigroups 2 , and split orthodox semigroups 6 . Several authors have i n vestigated regular semigroups with inverse transversals; see, for example, 1 , 4 , 11 , 12 . A general structure theorem for regular semigroups with inverse transversals was obtained by Saito 13 . In a regular semigroup with an inverse transversal S , the subsets I = fe 2 ES : e = ee g; = ff 2 ES : f = f fg;are of considerable importance. In this paper we show t h a t b o t h I and are subsemigroups of S. This means that, in the terminology of 13 , every inverse transversal of S is an S-inverse transversal, so that this latter concept becomes super uous. Consequently, the construction in 13 can be replaced by the simpler form given in 12 . We also obtain necessary and su cient conditions for an inverse subsemigroup T of S to be an inverse transversal, in particular when S is E-solidquasi-orthodox or locally inverse. We recall that for idempotents e, f of a regular semigroup S the sandwich set Se; f 8 is de ned by Se; f = fg 2 ES : ge= g = f g ;e g f= efg: It is well-known that Se; f = f Vefe. Moreover, if a 0 2 V a and b 0 2 V b, then, writing Sa; b for Sa 0 a; bb 0 ,w e h a ve 8g 2 Sa; b b 0 ga 0 2 V ab: In particular, if eLgRf with e; f; g 2 ES t h e n f g e= g and ef 2 V g.In a regular semigroup S, we list the following basic facts which will be used in the sequel:1. if aa 0 = a or a 0 a = a for a 0 2 V a t h e n a; a 0 2 ES a n d aLa 0 or aRa 0 ;

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