Linearly ordered sets with only one operator have the amalgamation property

TL;DR

This paper investigates the amalgamation properties of linearly ordered sets with various unary operations, revealing conditions under which these classes possess the strong amalgamation property and exploring related model-theoretic implications.

Contribution

It characterizes when classes of linearly ordered sets with unary operations have the amalgamation property, including the strong version, and extends results to order reversing operations.

Findings

Linearly ordered sets with one order-preserving unary operation have SAP.

Linearly ordered sets with one strict order-preserving unary operation have AP but not SAP.

Classes with multiple automorphisms lack AP.

Abstract

The class of linearly ordered sets with one order preserving unary operation has the Strong Amalgamation Property (SAP). The class of linearly ordered sets with one strict order preserving unary operation has AP but not SAP. The class of linearly ordered sets with two order preserving unary operations has not AP. For every set $F$, the class of linearly ordered sets with an $F$-indexed family of automorphisms has SAP. Corresponding results are proved in the case of order reversing operations. Various subclasses of the above classes are considered and some model-theoretical consequences are presented.