By Thomas W. Hungerford
Finally a self-contained, one quantity, graduate-level algebra textual content that's readable via the typical graduate scholar and versatile adequate to house a wide selection of teachers and direction contents. The guideline all through is that the cloth can be awarded as common as attainable, in step with strong pedagogy. consequently it stresses readability instead of brevity and includes a very huge variety of illustrative exercises.
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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!