On translations of complete first order theories

Abstract

In this paper, some basic properties of automorphisms of first order languages and some examples are presented. In particular, an omitting types theorem is proved.

Twists and Twistability

“…By Fact 5.13 we find that Γ is antipodal. Thus Γ does not realize the triangle type (5, 5, 2) and hence Γ σ does not realize the triangle type (3,3,2). This contradicts Proposition 7 of the Appendix.…”

“…By Fact 5.5, antipodal graphs contain no triangles of perimeter greater than 2δ. Thus Γ does not realize the triangle type (2,2,3). This disposes of the second point.…”

“…Now suppose that δ = 6. Since the triangle type (2,4,6) is of geodesic type, it must be realized in Γ σ (Observation 1), and therefore σ −1 (2, 4, 6) = (1, 2, σ −1 (6)) must be realized in Γ. This implies then that σ −1 (6) ≤ 3.…”

“…This gives us that σ(5) = 1, and thus σ = (14365). However, this permutation would send the geodesic type (2,3,5) to (1,2,6), and hence is not a suitable twist.…”

“…Twisted isomorphisms have been considered in a number of model theoretic contexts, namely, in [10,11,2,13,14,4], and [6]. In some of these articles, the authors refer to twisted isomorphisms as permorphisms.…”