By Neil Hindman; Dona Strauss
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Additional info for Algebra in the Stone-CМЊech compactification : theory and applications
2. 3 for which the smallest ideal exists. 3. Let S and T be semigroups and let h W S ! T be a surjective homomorphism. S/. If L is a minimal left ideal of S , show that hŒL is a minimal left ideal of T , and that the corresponding statement also holds for minimal right ideals. If S has a minimal left ideal which contains an idempotent, show that T does as well. p 0 / D p. 1]. The presentation of the Structure Theorem was suggested to us by J. Pym and is based on his treatment in . 64) is due to A.
Let S be a semigroup and assume that there is a minimal left ideal L of S which has an idempotent e. Then L D XG X G where X is the (left zero) semigroup of idempotents of L, and G D eL D eSe is a group. All maximal groups in L are isomorphic to G. Proof. Given x 2 L, Lx is a left ideal of S and Lx Â L so Lx D L and hence there is some y 2 L such that yx D e. 30 (b), e is a right identity for Le D L. 40 applies (with L replacing S). It is a routine exercise to show that the maximal groups of X G are the sets of the form ¹xº G.
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.
Algebra in the Stone-CМЊech compactification : theory and applications by Neil Hindman; Dona Strauss