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Extra resources for Algebra in the Stone-Cech compactification : Theory and Applications
Let L be a minimal left ideal of S and let J be a left ideal of S. Pick a 2 J . 46, La is a minimal left ideal which is contained in J . 48. S/. Statements (a) through (f) are equivalent and imply statement (g). If either S is simple or every left ideal of S has an idempotent, then all statements are equivalent. (a) Se is a minimal left ideal. (b) Se is left simple. (c) eSe is a group. e/. (e) eS is a minimal right ideal. (f) eS is right simple. (g) e is a minimal idempotent. 24 Chapter 1 Semigroups and Their Ideals Proof.
G/ is not the identity of F . Proof. Let n be the length of g, let X D ¹0; 1; : : : ; nº, and let F D ¹f 2 XX W f is one-to-one and onto Xº. F; ı/ is a group whose identity is Ã, the identity function from X to X . i 1/ D aº. a/ D ;. a/ ! a/ is one-to-one. a/ in any way to a member of F . Let b W G ! 22. k/ D k 1: To see this, first suppose that ik 1 D 1. k/ D k 1. Now suppose that ik 1 D 1. k/ D k 1. g/ is not the identity map. 1. 22. 3 Powers of a Single Element Suppose that x is a given element in a semigroup S.
Then xS is a right ideal , S x is a left ideal and SxS is an ideal. S/. Then e is a left identity for eS , a right identity for Se, and an identity for eSe. Proof. Statement (a) is immediate. S/. To see that e is a left identity for eS , let x 2 eS and pick t 2 S such that x D et . Then ex D eet D et D x. Likewise e is a right identity for Se. 31. Let S be a semigroup. S/, then e is a right identity for S. (b) If L is a left ideal of S and s 2 L, then Ss Â L. (c) Let ; ¤ L Â S . Then L is a minimal left ideal of S if and only if for each s 2 L, Ss D L.