Preservation of NATP

TL;DR

This paper establishes several preservation theorems for NATP, demonstrating that NATP is maintained under various theory expansions and constructions, and provides new examples including the model companion of ACFO.

Contribution

It introduces new preservation results for NATP under theory parametrization, sums, dense/co-dense expansions, and specific algebraic structures, along with new NATP examples.

Findings

NATP preserved under parametrization and sums of Fra"{i}ssé limit theories.

NATP preserved in dense/co-dense expansions like lovely pairs and H-structures.

The model companion of ACFO is proven to be NATP.

Abstract

We prove several preservation theorems for NATP and furnish several examples of NATP. First, we prove preservation of NATP for the parametrization and sum of the theories of Fra\"{i}ss\'{e} limits of Fra\"{i}ss\'{e} classes satisfying strong amalgamation property. Second, we prove preservation of NATP for two kinds of dense/co-dense expansions, that is, the theories of lovely pairs and of H-structures for geometric theories and dense/co-dense expansion on vector spaces. Third, we prove preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also preservations of NTP$_1$ and NTP$_2$ are considered. Fourth, we present some proper examples of NATP using the results proved in this paper. Most of all, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP.