By Isaacs I.M.

ISBN-10: 0821847996

ISBN-13: 9780821847992

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**Additional resources for Algebra. A graduate course**

**Example text**

Therefore, f £ C. (iii)=> (iv). S i n c e / ( / A 5 ) - 1 A g ( / A 9 ) - 1 = 1 for all / , g £ G, without loss of generality, c = / ( / A g)'1 £ C. Hence / = c(f A g) < eg. (iv) => (v). \D2 and d2 £ D$\DX. Without loss of generality, for some c £ C we have d\ < cd2. Thus d± £ D2 by convexity, a contradiction. (v) =* (i). A. By (v), C V A C C V B without loss of generality. // By (ii), if V is a value of g £ G\{1}, then V is a value of g+ or g~; thus values are always values of strictly positive elements.

9 In any lattice-ordered group, \g\ — 1 if and only if g = I. Hence (wi = 1 k. k wn = 1) i/ and only if\wi\ V ... V |tuB| = 1. g-n, then f = / i - . 9, ( l < j < n ) . Proof: By induction on n. -g„. fn for + 1 some f3 £ G with ft < g3 (1 < j < n). fn(f A gn+\)-ll 24 CHAPTER 2. 4). Suppose that G and H are groups that satisfy the same sentences of group theory. Then they have isomorphic ultrapowers (see Theorem LA). 29); so if G is the carrier of an ordered group, then so is H (restrict the order on the ultrapower of H to the subgroup order-isomorphic to H).

Consequently, h < g~l, so g~l = [\{g~l :i€ /}■ 22 CHAPTER 2. BASIC PROPERTIES (b) If ft. > fgt for all i e 7, then f~lh > gt for all i € I. Hence f~ h > 5, so ft > fg. Thus fg is the supremum of {fgt : i 6 / } . l (c) By (a) and (b), [\{gg~l : i € 7} exists and equals 1. 5r\ since g~\f/\g) = g~lf Al < ft X / A l = $f H / A f t ) . -1 Hence A{(/ A #)(/ A & ) : i e 7} exists and equals 1. The result now follows from (a) and (b). / Surprisingly, the above property can be captured by a first order sentence in the language of lattices.

### Algebra. A graduate course by Isaacs I.M.

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