By Neil Hindman; Dona Strauss
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Extra resources for Algebra in the Stone-CМЊech compactification : theory and applications
30 (b), s D se. Also, since S is simple SsS D S , so pick u and v in S with e D vsu. Let r D eue and t D ev. Then tsr D evseue D evsue D eee D e and er D eeue D eue D r. Let f D rt s. S/. Also, f e D rt se D rt s D f and ef D ert s D rt s D f so f Ä e so f D e. Thus Se D Sf D S rt s Â S s Â L. (c) Let L be a left ideal with L Â Se. We show that e 2 L (so that Se Â L and hence Se D L). Pick an idempotent t 2 L, and let f D et . Then f 2 L. Since t 2 Se, t D t e. Thus f D et D et e. S/. Also ef D eete D ete D f and f e D et ee D et e D f so f Ä e so f D e and hence e 2 L.
For each n 2 N, we define an element x n in S. We do this inductively, by stating that x 1 D x and that x nC1 D xx n if x n has already been defined. It is then straightforward to prove by induction that x m x n D x mCn for every m; n 2 N. Thus ¹x n W n 2 Nº is a commutative subsemigroup of S . We shall say that x has finite order if this subsemigroup is finite; otherwise we shall say that x has infinite order. If S has an identity e, we shall define x 0 for every x 2 S by stating that x 0 D e.
A) The principal ideal generated by x is S xS [ xS [ Sx [ ¹xº. (b) If S has an identity, then the principal ideal generated by x is SxS. (c) The principal left ideal generated by x is Sx [ ¹xº and the principal right ideal generated by x is xS [ ¹xº. Proof. 1. 1. 33. 2. Describe the ideals in each of the following semigroups. Also describe the minimal left ideals and the minimal right ideals in the cases in which these exist. N; C/. X /; [/, where X is any set. X /; \/, where X is any set. Œ0; 1; /, where denotes multiplication.
Algebra in the Stone-CМЊech compactification : theory and applications by Neil Hindman; Dona Strauss