By Neil Hindman; Dona Strauss

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3. Let X be any set. Describe the minimal left and right ideals in XX. 4. Let S be a commutative semigroup with an identity e. Prove that S has a proper ideal if and only if there is some s 2 S which has no e-inverse. In this case, prove that ¹s 2 S W s has no e-inverseº is the unique maximal proper ideal of S. 5 Chapter 1 Semigroups and Their Ideals Idempotents and Order Intimately related to the notions of minimal left and minimal right ideals is the notion of minimal idempotents. 34. S/. Then (a) eÄL f if and only if e D ef , (b) eÄR f if and only if e D f e, and (c) e Ä f if and only if e D ef D f e.

Observe, however, that many common semigroups do not have a smallest ideal. N; /. 52. Let S be a semigroup. (a) Let L be a left ideal of S . Then L is minimal if and only if Lx D L for every x 2 L. (b) Let I be an ideal of S. Then I is the smallest ideal if and only if I xI D I for every x 2 I . Proof. (a) If L is minimal and x 2 L, then Lx is a left ideal of S and Lx Â L so Lx D L. Now assume Lx D L for every x 2 L and let J be a left ideal of S with J Â L. Pick x 2 J . Then L D Lx Â LJ Â J Â L.

Let y be the inverse of x. Then xy 2 I so I D S. As promised earlier, we now see that any semigroup with a left identity e such that every element has a right e-inverse must be (isomorphic to) the Cartesian product of a group with a right zero semigroup. 40. Let S be a semigroup and let e be a left identity for S such that for each x 2 S there is some y 2 S with xy D e. S/ and let G D Se. Then Y is a right zero semigroup, G is a group, and S D GY G Y. Proof. We show first that: For all x 2 Y and for all y 2 S, xy D y.