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By [EDS] R.W. GATTERDAM AND K.W.WESTON

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Extra resources for Conference on Group Theory, University of Wisconsin-Parkside, 1972; [processing]

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The subsemigroup denned above is called the inverse subsemigroup gener­ ated by X and is denoted by (X). Clearly, the idempotents of an inverse semigroup form an inverse subsemi­ group. More generally, any inverse subsemigroup containing all the idempo­ tents is said to be full. The exact relationship between groups and inverse semigroups is spelt out by the following. Proposition 4 Groups are precisely the inverse semigroups with exactly one idempotent. Proof Clearly, groups are inverse semigroups having exactly one idempotent.

Proposition 3 Let X be a non-empty subset of an inverse semigroup S. Then the intersection of all inverse subsemigroups of S which contain X is an inverse subsemigroup of S. It consists precisely of all products of elements drawn from thesetXuX-1. ■ The subsemigroup denned above is called the inverse subsemigroup gener­ ated by X and is denoted by (X). Clearly, the idempotents of an inverse semigroup form an inverse subsemi­ group. More generally, any inverse subsemigroup containing all the idempo­ tents is said to be full.

For example, the structure of the principal right ideals turns out to be important in proving the Wagner-Preston theorem. 21 Abstract inverse semigroups Lemma 5 Let S be an inverse semigroup. (1) aS = aa~1S for all a £ S, and aa"1 is the unique idempotent generator of aS (2) Sa = Sa~la for all a £ S, and a~la is the unique idempotent generator of Sa. (3) eS fl fS = efS where e and f are idempotents. (4) Se fl Sf = Sef where e and f are idempotents. Proof (1) We have that aS = (aa~l)aS C aa^S C oS.