Infinite Words and Nonstandard Completions of Finite Pregroups

Let Λ be a discretly ordered abelian group. In that event the positive class N of the ordred abelian group Z embeds in the positve class Λ+ of Λ as an initial segment. If A is any nonempty set and λ is any positive element of Λ, then by an infinite word (of length λ) on the alphabet A is meant a function from the intial segment [1, λ] = {x ∈ Λ : 1 ≤ x ≤ λ} of Λ+ into A. A.G. Myasnikov suggested to the authors that they explore completions of pregroups P using infinite words. We show how this can be done if we take an excursion to a nonstandard extension ∗Z of Z and replace Λ with a subgroup Λ ≤ Λ ≤∗ Z convex in ∗Z. We restrict ourselves to finite pregroups P for, in the contrary case, we could have as entries in our Λ-words not 26 Anthony M. Gaglione, Seymour Lipschutz and Dennis Spellman only individual letters from P but also formal infinite products in the corresponding nonstndard extension ∗P of P . Mathematics Subject Classification: Primary 20E06, 03C03, 03H03