By A. M. W. Glass, W. Charles Holland (auth.), A. M. W. Glass, W. Charles Holland (eds.)

ISBN-10: 9400922833

ISBN-13: 9789400922839

ISBN-10: 9401075247

ISBN-13: 9789401075244

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**Extra info for Lattice-Ordered Groups: Advances and Techniques**

**Example text**

Algorithmically. For a given set 4> of formulas, let Th<1>(~' the tP-theory of denote the set of all sentences in 4> that hold in all i-groups G E '1(; in particular, for 4>=SE,Th(1() = Th<1>(1() denotes the (elementary) theory of 'l( Th<1>(1() is decidable if there is an algorithm which, on input of a sentence q>E <1>, decides whether or not q> is valid in all GE 'l( Thus, in principle, a decidable theory is trivial. 5tr of all Archimedean i -groups is hereditarily undecidable. 5tr . in particular for the class of all i-groups - Th(~ is undecidable.

6, [CvG(f)lI\[CvG(g)] = CV[G(f)I\G(g)) = CVG(fl\g) = C. , fe Cor gE C. (iv) => (i). f(fl\g)-ll\g(fl\g)-l = e for all f,gE G. So, without loss of generality, f(fl\g)-l E C. Hence Cf = C(fl\g) S Cg. Therefore C is prime. 7. The set of convex i-subgroups of an i-group G is linearly ordered by inclusion if and only if G is an o-group. 14 Chapter 1 There is one way to obtain prime subgroups that will prove very useful. Let C be a convex i-subgroup of an i-group G and ge G\C. By Zorn's Lemma, there is a convex i-subgroup V of G containing C, maximal with respect to the property that g~ V.

F3n with a < f31 < f32 < ... 1. Hence we can find XE A(n) such that ax = f3I' and f3iX = f3i+I' Taking x as defined only on the interval [a,f3n-I]' we can extend its action to [f3n-I,f3n] by letting x =xn-Ig. We continue in this fashion both to the right and to the left, so that on the entire supporting interval, x n =g. Any conjugate of a bump is a bump of the same type. Moreover, the bump structure of every member of A(n) is preserved by conjugation, in the following sense. For each IE A(n) let B(f) be the set of all pairs (I ,a) such that I is a fixed point of I and (J.

### Lattice-Ordered Groups: Advances and Techniques by A. M. W. Glass, W. Charles Holland (auth.), A. M. W. Glass, W. Charles Holland (eds.)

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