By M. A. Krasnosel'skii, Yz. B. Rtuickii
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1. SEMIGROUPS, MONOIDS AND GROUPS 25 If G is a monoid and the binary operation is written multiplicatively, then the identity element of G will always be denoted e. If the binary operation is written additively, then a + h (a, beG) is called the sum of a and b, and the identity element is denoted 0; if G is a group the inverse of a eGis denoted by -a. We write a - b for a + (-h). Abelian groups are frequently written additively. 1 to define a group can actually be weakened considerably. 3. Let G be a semigroup.
An e G, then for any permutation it, ... , in of 1, 2, ... n, ala2·· ·a n = ailai2· .. ai n· PROOF. Exercise. 8. Let G be a semigroup, a EG andn e N*. The element an e G is defined n to be the standard n product IT ai with ai = i=1 a for 1 ~ i ~ n. I/G is a monoid, aD is defined to be the identity element e. I/G is a group, then for each n E N*, a- n is defined to be (a-I)n e G. 6 and Exercise 16 show that exponentiation is well defined. By definition, then, a l = a, a2 = aa, a3 = (aa)a = aaa, ...
7) and verify that the hypotheses of Zorn's Lemma are satisfied. e 0. To do this note that the map N X 10,1) ---+ N given by (n,O) f-> 2n and (n, 1) f-> 2n 1 is a bijection. 8). Therefore by Zorn's Lemma there is a maximal element (g,C) e 5'. + + + + Clearly Co = I(c,O) Ice C) and Cl ICol = let = ICd and Cx 10,1) = Co jection. 3, let = IC X 10,1)1 = ICo = I (c,l) Ice C) are disjoint sets such that U C1• Themapg:CX 10,1)---+Cisa bi- U Cd = ICol + ICll = let + let· 20 PREREQUISITES AND PRELIMINARIES To complete the proof we shall show that Ie!