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]
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.