By Neil Hindman; Dona Strauss
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Additional resources for Algebra in the Stone-CМЊech compactification : theory and applications
A) There is a minimal left ideal of L which has an idempotent. (b) There is a minimal left ideal of L \ R which has an idempotent. In fact if J is a minimal left ideal of S with J Â L, then R \ J is a minimal left ideal of R \ L which has an idempotent. (c) There is a minimal left ideal of R which has an idempotent. Proof. 47 a minimal left ideal J of S such that J Â L. 43 (b), I D J . 56, J has an idempotent. 47 a minimal right ideal M of S such that M Â 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 Â R \ J .
X; y/ 2 X Y . 42. 42. 62 all minimal left ideals of S are isomorphic and by (e) X G ¹vº is isomorphic to X G. The other conclusion follows similarly. y; x/ 2 Y X . 63 (g) we have that for any y 2 Y , X G ¹yº X G. However, there is no transparent isomorphism unless Œy; x D e for all x 2 X, such as when y D v. 63 spells out in detail the structure of X G Y . We see now that this is in fact the structure of the smallest ideal of any semigroup which has a minimal left ideal with an idempotent. 64 (The Structure Theorem).
B) S is both left simple and right simple. 20 Chapter 1 Semigroups and Their Ideals (c) For all a and b in S, the equations ax D b and ya D b have solutions x; y in S. (d) S is a group. Proof. (a) implies (b). Pick an idempotent e in S. We show first that e is a (two sided) identity for S. Let x 2 S . Then ex D eex so by left cancellation x D ex. Similarly, x D xe. To see that S is left simple, let L be a left ideal of S. Then LS is an ideal of S so LS D S, so pick t 2 L and s 2 S such that e D t s.
Algebra in the Stone-CМЊech compactification : theory and applications by Neil Hindman; Dona Strauss