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By Neil Hindman; Dona Strauss

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Given y; z 2 C one has yz 2 AA Â A and yzx D yx D x so yz 2 C . Thus C 2 A. Since C Â A and A is minimal, we have C D A so x 2 C and so xx D x as required. ” Because of the following corollary, we are able to incorporate all of these results. 6. Let S be a compact right topological semigroup. Then every left ideal of S contains a minimal left ideal. Minimal left ideals are closed, and each minimal left ideal has an idempotent. Proof. If L is any left ideal L of S and x 2 L, then Sx is a compact left ideal contained in L.

XX; V /. x// in X . XX ; V /, and so f is continuous. This establishes (a). XX; V / and every x 2 X . This is obviously the case if f is continuous. Conversely, suppose that f is continuous. Let hxà iÃ2I be a net converging to x in X. We define gà D xà , the function in XX which is constantly equal to xà and g D x. Then hgà iÃ2I converges to g in XX and so hf ı gà iÃ2I converges to f ı g. x/. Thus f is continuous, and we have established (b). 3. Let X be a topological space. The following statements are equivalent: (a) XX is a topological semigroup.

47 a minimal left ideal J of S such that J  L. 43 (b), I D J . 56, J has an idempotent. 47 a minimal right ideal M of S such that M  R. 61 an idempotent e 2 M \ J . To see that R \ J is a minimal left ideal of R \ L, let I be a left ideal of R \ L with I  R \ J . To see that R \ J  I , let u 2 R \ J . Pick x 2 I . 52 (a) so pick y 2 J such that e D yx. Then e D ee D eyx. R \ L/x  I so e 2 I . Now u 2 J D Je so u D ue. R \ L/e  I as required. (c) By (b) R \ J is a minimal left ideal of R \ L.

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