By Neil Hindman; Dona Strauss

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X/ D x for every x 2 f ŒX . 12 Chapter 1 Semigroups and Their Ideals We next define the concept of a free group on a given set of generators. The underlying idea is simple, but the rigorous definition may seem a little troublesome. The basic idea is that we want to construct all expressions of the form a1e1 a2e2 akek , where each ai 2 A and each exponent ei 2 Z; and to combine them in the way that we are forced to by the group axioms. 21. a; i /. b; j /. If k D m D n, then f g D ;. G; / is the free group generated by A.

K; r/ 2 S W k > mº is a right ideal of S which is properly contained in R. 55. 1. 45. 2. 52 (b). 3. Let S D ¹f 2 NN W f is one-to-one and N nf ŒN is infiniteº. S; ı/ is left simple (so S is a minimal left ideal of S) and S has no idempotents. 4. Suppose that a minimal left ideal L of a semigroup is commutative. Prove that L is a group. 5. Let S be a semigroup and assume that there is a minimal left ideal of S. S / is commutative, then it is a group. 7 Minimal Left Ideals with Idempotents We present here several results that have as hypothesis “Let S be a semigroup and assume that there is a minimal left ideal of S which has an idempotent”.

D) implies (a). S/ ¤ ;. To see that S is simple, let I be an ideal of S and pick x 2 I . 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.